02-Algebra and Trigonometry...॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर:...
Transcript of 02-Algebra and Trigonometry...॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर:...
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
ALGEBRA AND TRIGONOMETRY
Dr.N.A.Pande Associate Professor
Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded – 431602
Maharashtra, INDIA
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
PAPER DETAILS
2
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
PAPER DETAILS
2
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
• Semester : 1st
PAPER DETAILS
2
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
• Semester : 1st
• Paper No. : 02(B.A.) / 02(B.Sc.)
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
• Semester : 1st
• Paper No. : 02(B.A.) / 02(B.Sc.)
• Syllabus Effective From : 2016-17
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
• Semester : 1st
• Paper No. : 02(B.A.) / 02(B.Sc.)
• Syllabus Effective From : 2016-17
• Paper Code : CCM-1 Section : B
PAPER DETAILS
2
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 1st
• Semester : 1st
• Paper No. : 02(B.A.) / 02(B.Sc.)
• Syllabus Effective From : 2016-17
• Paper Code : CCM-1 Section : B
• Marks : 40 (University) + 10 (Internal) = 50
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
• Matrices : Matrix, Different Types of Matrices, Equality of Matrices, Addition (Sum) of Two Matrices, Properties of Matrix Addition, Subtraction of Two Matrices, Multiplication of a Matrix by a Scalar, Properties of Multiplication of a Matrix by a Scalar, Multiplication of Two Matrices, Properties of Matrix Multiplication, Positive Integral Powers of a Matrix, Transpose of a Matrix, Conjugate of a Matrix, Transposed Conjugate of a Matrix, Determinant of a Square Matrix, Minor of an Element, Co-factor of an Element, Adjoint of a Square Matrix, Inverse of a Square Matrix, Singular and Non-singular Matrix, Orthogonal Matrices, The Determinant of an Orthogonal Matrix, Unitary Matrix.
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-II
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-II
• Rank of a Matrix and Linear Equations : Minor of Order k of a Matrix, Rank of a Matrix, Elementary Row and Column Operations, Elementary Operations, The Inverse of an Elementary Operation, Row and Column Equivalent, Equivalent Matrices, Working Procedure for Finding Rank Using Elementary Operations, Row-Echelon Matrix, Row Rank and Column Rank of a Matrix, Linear Equations, Equivalent Systems, System of Homogeneous Equations.
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-II
• Rank of a Matrix and Linear Equations : Minor of Order k of a Matrix, Rank of a Matrix, Elementary Row and Column Operations, Elementary Operations, The Inverse of an Elementary Operation, Row and Column Equivalent, Equivalent Matrices, Working Procedure for Finding Rank Using Elementary Operations, Row-Echelon Matrix, Row Rank and Column Rank of a Matrix, Linear Equations, Equivalent Systems, System of Homogeneous Equations.
• Characteristic Roots and Characteristic Vectors : Definitions, To Find Characteristic Vectors, Cayley-Hamilton Theorem (Statement Only)
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-III
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-III
• Trigonometry : Complex Quantities, DeMoivre’s Theorem, Expansions of sin nθ and cos nθ, Expansions of the sine and cosine of an Angle in Series of Ascending Powers of the Angle, Expansions of the sines and cosines of Multiple Angles, and of Powers of sines and cosines, Exponential Series for Complex Quantities, Circular Functions for Complex Angles, Hyperbolic Functions, Inverse Circular Functions, Inverse Hyperbolic Functions.
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
PAPER DETAILS
6
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
PAPER DETAILS
6
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
PAPER DETAILS
6
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
�Edition : First Edition, 1997
PAPER DETAILS
6
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
�Edition : First Edition, 1997
• Scope :
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
�Edition : First Edition, 1997
• Scope : • Unit–I : Chapter 10 : 10.1 to 10.17 (10.13, 10.15,
10.17 Only Statements), 10.20 to 10.22, 10.27 to 10.32, 10.34 to 39 (10.39 Only Statements)
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
�Edition : First Edition, 1997
• Scope : • Unit–I : Chapter 10 : 10.1 to 10.17 (10.13, 10.15,
10.17 Only Statements), 10.20 to 10.22, 10.27 to 10.32, 10.34 to 39 (10.39 Only Statements)
Unit–II : Chapter 11 : 11.1, 11.2, 11.5 to 11.16, 11.32 to 11.39
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Authors : Om P.Chug, K.Prakash, A.D.Gupta
�Publisher : Anmol Publications, New Delhi
�Edition : First Edition, 1997
• Scope : • Unit–I : Chapter 10 : 10.1 to 10.17 (10.13, 10.15,
10.17 Only Statements), 10.20 to 10.22, 10.27 to 10.32, 10.34 to 39 (10.39 Only Statements)
Unit–II : Chapter 11 : 11.1, 11.2, 11.5 to 11.16, 11.32 to 11.39
• Chapter 12 : 12.1 to 12.3, 12.18 (Only Statement)
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
PAPER DETAILS
7
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
�Authors : S.L.Loney
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
�Authors : S.L.Loney
�Publisher : A.I.T.B.S. Publishers and Distributors, Delhi
PAPER DETAILS
7
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
�Authors : S.L.Loney
�Publisher : A.I.T.B.S. Publishers and Distributors, Delhi
�Edition : Reprint, 2003
PAPER DETAILS
7
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
�Authors : S.L.Loney
�Publisher : A.I.T.B.S. Publishers and Distributors, Delhi
�Edition : Reprint, 2003
• Scope :
PAPER DETAILS
7
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Plane Trigonometry Part II
�Authors : S.L.Loney
�Publisher : A.I.T.B.S. Publishers and Distributors, Delhi
�Edition : Reprint, 2003
• Scope : • Unit–III : Art. 17, 18 19, 21, 22, 27, 32, 33, 42, 43, 44,
45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 67, 68, 69, 71, 73, 74, 76, 77, 79.
PAPER DETAILS
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Reference Books
PAPER DETAILS
8
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Reference Books • A Text Book of Matrices By Shanti Narayan (S.Chand & Company Ltd., New
Delhi)
• Matrices By A.R.Vasishtha (Krishna Prakashan Media (P) Ltd., Meerut)
• First Course in Linear Algebra by P.B.Bhattacharya, S.K.Jain, S.R.Nagpaul (New Age International (P) Limited Publishers)
• Elementary Topics in Algebra By K. Khurana and S.B. Malik. (Vikas Publishing House Pvt. Ltd., New Delhi.)
• Higher Trigonometry B. C. Das, B. N. Mukherjee, By (U.N.Dhur & Sons Private Ltd. Kolkata)
• Arihant Trigonometry, Amit M. Agrawal (Arihant Publication Pvt. Ltd).
• Lectures on Algebra and Trigonometry By T M Karade and M S Bendre, Sonu Nilu Bandu, Nagpur.
• Text Book on Trigonometry By R S Verma and K. S. Shukla, Pothishala Private limited pub.
• Elementry Matrix Algebra By Hohn Franz E, Amerind Pub. Co. Pvt. Ltd.
• Text Book on Algebra and Theory of Equations By Chandrika Prasad, Pothishala Private limited pub.
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • Matrix is an arrangement of mn numbers in m rows and n columns enclosed in brackets.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • Matrix is an arrangement of mn numbers in m rows and n columns enclosed in brackets.
• Matrices are denoted by upper case letters like A, B, C etc.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • Matrix is an arrangement of mn numbers in m rows and n columns enclosed in brackets.
• Matrices are denoted by upper case letters like A, B, C etc.
• Elements of matrices are denoted by corresponding lower case letters with two indices like aij, bij, cij where first index i is the row number and the second index j is the column number in which the element is present.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
�upper triangular matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
�upper triangular matrix
� lower triangular matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
�upper triangular matrix
� lower triangular matrix
�diagonal matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
�upper triangular matrix
� lower triangular matrix
�diagonal matrix
� scalar matrix
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Different Types of Matrices • There are different types of matrices
�Zero or null matrix (denoted by O)
� row matrix
�column matrix
� rectangular matrix
� square matrix
� singular matrix
�upper triangular matrix
� lower triangular matrix
�diagonal matrix
� scalar matrix
� Identity / unit matrix (denoted by I)
UNIT-I
10
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Equality and Addition of Matrices
UNIT-I
11
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Equality and Addition of Matrices • Equality of Two Matrices : For A = [aij]m×n
and B = [bij]p×q, A = B ⇔ m = p, n = q and aij = bij ∀ i, j.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Equality and Addition of Matrices • Equality of Two Matrices : For A = [aij]m×n
and B = [bij]p×q, A = B ⇔ m = p, n = q and aij = bij ∀ i, j.
• Addition or Sum of Matrices : Two matrices can be added ⇔ they are of same order and in this case they are said to be conformable for addition.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Equality and Addition of Matrices • Equality of Two Matrices : For A = [aij]m×n
and B = [bij]p×q, A = B ⇔ m = p, n = q and aij = bij ∀ i, j.
• Addition or Sum of Matrices : Two matrices can be added ⇔ they are of same order and in this case they are said to be conformable for addition.
• If A = [aij]m×n and B = [bij]m×n, then A + B = [aij + bij]m×n
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
� If A, B and C are matrices of same order, then
(A + B) + C = A + (B + C)
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
� If A, B and C are matrices of same order, then
(A + B) + C = A + (B + C)
• Zero matrix is identity for matrix addition.
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
� If A, B and C are matrices of same order, then
(A + B) + C = A + (B + C)
• Zero matrix is identity for matrix addition.
� If A and O are matrices of same order, then
A + O = O + A = A
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
� If A, B and C are matrices of same order, then
(A + B) + C = A + (B + C)
• Zero matrix is identity for matrix addition.
� If A and O are matrices of same order, then
A + O = O + A = A
• Negative of matrix is its additive inverse.
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Addition • Matrix addition is commutative.
� If A and B are matrices of same order, then
A + B = B + A.
• Matrix addition is associative.
� If A, B and C are matrices of same order, then
(A + B) + C = A + (B + C)
• Zero matrix is identity for matrix addition.
� If A and O are matrices of same order, then
A + O = O + A = A
• Negative of matrix is its additive inverse.
�For any matrix A, A + (−A) = (−A) + A = O
UNIT-I
12
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
• A − B = A + (−B)
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
• A − B = A + (−B)
• Matrix subtraction is not commutative.
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
• A − B = A + (−B)
• Matrix subtraction is not commutative.
A − B ≠ B − A
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
• A − B = A + (−B)
• Matrix subtraction is not commutative.
A − B ≠ B − A
• Matrix subtraction is not associative.
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Subtraction of Matrices • Subtraction of Matrices : A matrix can be
subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.
• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n
• A − B = A + (−B)
• Matrix subtraction is not commutative.
A − B ≠ B − A
• Matrix subtraction is not associative.
(A − B) − C ≠ A − (B − C)
UNIT-I
13
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
• Properties of Scalar Multiplication
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
• Properties of Scalar Multiplication
• x(A + B) = xA + xB
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
• Properties of Scalar Multiplication
• x(A + B) = xA + xB
• (x + y)A = xA + yA
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
• Properties of Scalar Multiplication
• x(A + B) = xA + xB
• (x + y)A = xA + yA
• (xy)A = x(yA)
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix
and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.
• Properties of Scalar Multiplication
• x(A + B) = xA + xB
• (x + y)A = xA + yA
• (xy)A = x(yA)
• 1A = A
UNIT-I
14
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrices
UNIT-I
15
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrices • Multiplication of Matrices : Two matrices
can be multiplied ⇔ number of columns in first is equal to number of rows in second and in this case they are said to be conformable for multiplication.
UNIT-I
15
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrices • Multiplication of Matrices : Two matrices
can be multiplied ⇔ number of columns in first is equal to number of rows in second and in this case they are said to be conformable for multiplication.
• A = [aij]m×n and B = [bij]n×p, AB = [cij]m×p, where .
UNIT-I
15
1
n
ij ik kj
k
c a b====
==== ∑∑∑∑
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrices • Multiplication of Matrices : Two matrices
can be multiplied ⇔ number of columns in first is equal to number of rows in second and in this case they are said to be conformable for multiplication.
• A = [aij]m×n and B = [bij]n×p, AB = [cij]m×p, where .
• Matrix multiplication is associative.
UNIT-I
15
1
n
ij ik kj
k
c a b====
==== ∑∑∑∑
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Multiplication of Matrices • Multiplication of Matrices : Two matrices
can be multiplied ⇔ number of columns in first is equal to number of rows in second and in this case they are said to be conformable for multiplication.
• A = [aij]m×n and B = [bij]n×p, AB = [cij]m×p, where .
• Matrix multiplication is associative.
• Matrix multiplication is distributive over matrix addition
UNIT-I
15
1
n
ij ik kj
k
c a b====
==== ∑∑∑∑
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
�BA is defined but is of different order than AB or
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
�BA is defined but is of different order than AB or
�BA is defined, is of order of AB but not equal to AB or
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
�BA is defined but is of different order than AB or
�BA is defined, is of order of AB but not equal to AB or
� (very rarely) BA is equal to AB.
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
�BA is defined but is of different order than AB or
�BA is defined, is of order of AB but not equal to AB or
� (very rarely) BA is equal to AB.
• AB = 0 may not guarantee A = 0 or B = 0.
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Matrix Multiplication • Matrix multiplication is not commutative.
• When AB is defined, about BA there are four cases, viz.,
�BA is just not defined or
�BA is defined but is of different order than AB or
�BA is defined, is of order of AB but not equal to AB or
� (very rarely) BA is equal to AB.
• AB = 0 may not guarantee A = 0 or B = 0.
• AB = AC may not guarantee B = C.
UNIT-I
16
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix
UNIT-I
17
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
UNIT-I
17
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
UNIT-I
17
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
• A1 = A
UNIT-I
17
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
• A1 = A
• A2 = AA
UNIT-I
17
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
• A1 = A
• A2 = AA
UNIT-I
17
times
n
n
A AA A==== ⋯⋯⋯⋯��������������������
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
• A1 = A
• A2 = AA
• AmAn = Am+n
UNIT-I
17
times
n
n
A AA A==== ⋯⋯⋯⋯��������������������
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A
is a square matrix, then
• A0 = I
• A1 = A
• A2 = AA
• AmAn = Am+n
• (Am)n = Amn
UNIT-I
17
times
n
n
A AA A==== ⋯⋯⋯⋯��������������������
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
• (A’)’ = A
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
• (A’)’ = A
• (kA)’ = kA’
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
• (A’)’ = A
• (kA)’ = kA’
• (A ± B)’ = A’ ± B’
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
• (A’)’ = A
• (kA)’ = kA’
• (A ± B)’ = A’ ± B’
• (AB)’ = B’A’
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,
then transpose of A is A’ = [aji]n×m.
• Properties of transpose of a matrix :
• (A’)’ = A
• (kA)’ = kA’
• (A ± B)’ = A’ ± B’
• (AB)’ = B’A’
• (An)’ = (A’)n
UNIT-I
18
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix
UNIT-I
19
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
UNIT-I
19
ijm n
A a××××
====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
kA kA====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
kA kA====
( )A B A B± = ±± = ±± = ±± = ±
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
kA kA====
( )A B A B± = ±± = ±± = ±± = ±
( )AB AB====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
kA kA====
( )A B A B± = ±± = ±± = ±± = ±
( )AB AB====
(((( )))) ( )n
nA A====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,
then conjugate of .
• Properties of conjugate of a matrix :
UNIT-I
19
ijm n
A a××××
====
A A====
kA kA====
( )A B A B± = ±± = ±± = ±± = ±
( )AB AB====
(((( )))) ( )n
nA A====''A A====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix
UNIT-I
20
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
UNIT-I
20
jim n
A aθθθθ××××
====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
• (Aθ)θ = A
UNIT-I
20
jim n
A aθθθθ××××
====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
• (Aθ)θ = A
UNIT-I
20
jim n
A aθθθθ××××
====
(((( ))))kA kAθθθθ θθθθ====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
• (Aθ)θ = A
• (A ± B)θ = Aθ ± Bθ
UNIT-I
20
jim n
A aθθθθ××××
====
(((( ))))kA kAθθθθ θθθθ====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
• (Aθ)θ = A
• (A ± B)θ = Aθ ± Bθ
• (AB) θ = BθAθ
UNIT-I
20
jim n
A aθθθθ××××
====
(((( ))))kA kAθθθθ θθθθ====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is
• (Aθ)θ = A
• (A ± B)θ = Aθ ± Bθ
• (AB) θ = BθAθ
• (An)θ = (Aθ)n
UNIT-I
20
jim n
A aθθθθ××××
====
(((( ))))kA kAθθθθ θθθθ====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Determinant of a Square Matrix
UNIT-I
21
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Determinant of a Square Matrix • Determinant of a Square Matrix : If A = [aij]n×n is a square matrix, then
UNIT-I
21
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Determinant of a Square Matrix • Determinant of a Square Matrix : If A = [aij]n×n is a square matrix, then
If n = 1, det A = a11.
UNIT-I
21
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Determinant of a Square Matrix • Determinant of a Square Matrix : If A = [aij]n×n is a square matrix, then
If n = 1, det A = a11.
If n > 1
UNIT-I
21
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Determinant of a Square Matrix • Determinant of a Square Matrix : If A = [aij]n×n is a square matrix, then
If n = 1, det A = a11.
If n > 1
where det Aij is the determinant of square submatrix of A obtained by deleting ith row and jth column.
UNIT-I
21
1
| | det ( 1) detn
i j
ij ij
i
A A a A++++
====
= = −= = −= = −= = −∑∑∑∑
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor & Cofactor of an Element
UNIT-I
22
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor & Cofactor of an Element • If A is a square a matrix, then Mij = det Aij,
which is the determinant of square submatrix of A obtained by deleting ith row and jth column is called minor of element aij.
UNIT-I
22
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor & Cofactor of an Element • If A is a square a matrix, then Mij = det Aij,
which is the determinant of square submatrix of A obtained by deleting ith row and jth column is called minor of element aij.
• If A is a square a matrix, then Cij = (−1)i+jdet Aij is the cofactor of element aij.
UNIT-I
22
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor & Cofactor of an Element • If A is a square a matrix, then Mij = det Aij,
which is the determinant of square submatrix of A obtained by deleting ith row and jth column is called minor of element aij.
• If A is a square a matrix, then Cij = (−1)i+jdet Aij is the cofactor of element aij.
• So, Cij = (−1)i+jMij.
UNIT-I
22
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor & Cofactor of an Element • If A is a square a matrix, then Mij = det Aij,
which is the determinant of square submatrix of A obtained by deleting ith row and jth column is called minor of element aij.
• If A is a square a matrix, then Cij = (−1)i+jdet Aij is the cofactor of element aij.
• So, Cij = (−1)i+jMij.
UNIT-I
22
1 1 1
| | ( 1) det ( 1)n n n
i j i j
ij ij ij ij ij ij
i i i
A a A a M a C+ ++ ++ ++ +
= = == = == = == = =
= − = − == − = − == − = − == − = − =∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
• A(adj A) = (adj A)A = |A|In.
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
• A(adj A) = (adj A)A = |A|In.
• |adj A| = |A|n−1, if |A| ≠ 0.
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
• A(adj A) = (adj A)A = |A|In.
• |adj A| = |A|n−1, if |A| ≠ 0.
• adj A’ = (adj A)’
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
• A(adj A) = (adj A)A = |A|In.
• |adj A| = |A|n−1, if |A| ≠ 0.
• adj A’ = (adj A)’
• Adjoint of a unit matrix is unit matrix.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n
is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.
• A(adj A) = (adj A)A = |A|In.
• |adj A| = |A|n−1, if |A| ≠ 0.
• adj A’ = (adj A)’
• Adjoint of a unit matrix is unit matrix.
• Adjoint of a symmetric matrix is symmetric matrix.
UNIT-I
23
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
UNIT-I
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
• A has inverse ⇔ |A| ≠ 0
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
• A has inverse ⇔ |A| ≠ 0
• A−1 = (adj A)/|A|
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
• A has inverse ⇔ |A| ≠ 0
• A−1 = (adj A)/|A|
• (A−1)−1 = A
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
• A has inverse ⇔ |A| ≠ 0
• A−1 = (adj A)/|A|
• (A−1)−1 = A
• (AB)−1 = B−1A−1
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix
: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.
• Inverse of A is denoted by A−1.
• Inverse of a matrix, when it exists, is unique.
• A has inverse ⇔ |A| ≠ 0
• A−1 = (adj A)/|A|
• (A−1)−1 = A
• (AB)−1 = B−1A−1
• (A’)−1 = (A−1)’
UNIT-I
24
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Singular and Non-singular Matrix
UNIT-I
25
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Singular and Non-singular Matrix • If A is a square matrix, the we can determine
its determinant.
UNIT-I
25
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Singular and Non-singular Matrix • If A is a square matrix, the we can determine
its determinant.
• A square matrix A is called singular matrix if, and only if, if |A| = 0.
UNIT-I
25
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Singular and Non-singular Matrix • If A is a square matrix, the we can determine
its determinant.
• A square matrix A is called singular matrix if, and only if, if |A| = 0.
• A square matrix A is called non-singular matrix if, and only if, if |A| ≠ 0.
UNIT-I
25
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Singular and Non-singular Matrix • If A is a square matrix, the we can determine
its determinant.
• A square matrix A is called singular matrix if, and only if, if |A| = 0.
• A square matrix A is called non-singular matrix if, and only if, if |A| ≠ 0.
• Every square matrix is either singular or non-singular.
UNIT-I
25
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
• For orthogonal matrix A’ = A−1.
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
• For orthogonal matrix A’ = A−1.
• Determinant of orthogonal matrix is ±1 and accordingly the orthogonal matrix is proper or improper.
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
• For orthogonal matrix A’ = A−1.
• Determinant of orthogonal matrix is ±1 and accordingly the orthogonal matrix is proper or improper.
• Inverse of orthogonal matrix is orthogonal
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
• For orthogonal matrix A’ = A−1.
• Determinant of orthogonal matrix is ±1 and accordingly the orthogonal matrix is proper or improper.
• Inverse of orthogonal matrix is orthogonal
• Transpose of orthogonal matrix is orthogonal
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.
• For orthogonal matrix A’ = A−1.
• Determinant of orthogonal matrix is ±1 and accordingly the orthogonal matrix is proper or improper.
• Inverse of orthogonal matrix is orthogonal
• Transpose of orthogonal matrix is orthogonal
• Product of orthogonal matrices is orthogonal
UNIT-I
26
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
• Inverse of unitary matrix is unitary.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
• Inverse of unitary matrix is unitary.
• Transpose of unitary matrix is unitary.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
• Inverse of unitary matrix is unitary.
• Transpose of unitary matrix is unitary.
• Conjugate of unitary matrix is unitary.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
• Inverse of unitary matrix is unitary.
• Transpose of unitary matrix is unitary.
• Conjugate of unitary matrix is unitary.
• Transposed conjugate of unitary matrix is unitary.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.
• For unitary matrix Aθ = A−1.
• Determinant of unitary matrix has absolute value 1.
• Inverse of unitary matrix is unitary.
• Transpose of unitary matrix is unitary.
• Conjugate of unitary matrix is unitary.
• Transposed conjugate of unitary matrix is unitary.
• Product of unitary matrices is unitary.
UNIT-I
27
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor of Order k of a Matrix
UNIT-II
28
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor of Order k of a Matrix • If A any matrix of order m × n, then for any k ≤ min{m, n}, minor of order k of A is determinant of a submatrix of order k × k of A.
UNIT-II
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Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor of Order k of a Matrix • If A any matrix of order m × n, then for any k ≤ min{m, n}, minor of order k of A is determinant of a submatrix of order k × k of A.
• Minor of any matrix can be found, irrespective of the original matrix is square or not.
UNIT-II
28
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor of Order k of a Matrix • If A any matrix of order m × n, then for any k ≤ min{m, n}, minor of order k of A is determinant of a submatrix of order k × k of A.
• Minor of any matrix can be found, irrespective of the original matrix is square or not.
• A matrix has many minors, of various orders.
UNIT-II
28
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Minor of Order k of a Matrix • If A any matrix of order m × n, then for any k ≤ min{m, n}, minor of order k of A is determinant of a submatrix of order k × k of A.
• Minor of any matrix can be found, irrespective of the original matrix is square or not.
• A matrix has many minors, of various orders.
• The orders of minors of a matrix range from 1 to min{m, n}.
UNIT-II
28
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.
• ρ(A) = 0 ⇔ A is a zero matrix
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.
• ρ(A) = 0 ⇔ A is a zero matrix
• ρ(A) ≤ min{m, n}
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.
• ρ(A) = 0 ⇔ A is a zero matrix
• ρ(A) ≤ min{m, n}
• A has a non-zero minor of rank k ⇒ ρ(A) ≥ k
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.
• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.
• ρ(A) = 0 ⇔ A is a zero matrix
• ρ(A) ≤ min{m, n}
• A has a non-zero minor of rank k ⇒ ρ(A) ≥ k
• Every minor of A of order k is 0 ⇒ ρ(A) < k
UNIT-II
29
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
�denoted by Ri,j.
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
�denoted by Ri,j.
• Multiplication to all elements of any one row (ith) by a fixed non-zero scalar (λ).
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
�denoted by Ri,j.
• Multiplication to all elements of any one row (ith) by a fixed non-zero scalar (λ).
�denoted by Ri(λ).
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
�denoted by Ri,j.
• Multiplication to all elements of any one row (ith) by a fixed non-zero scalar (λ).
�denoted by Ri(λ).
• Addition to all elements of any one row (ith) a fixed scalar (λ) multiple of corresponding elements of another row (jth).
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Row Operations • There are three types of elementary row
operations or transformations :
• Interchange of any two (ith and jth) rows.
�denoted by Ri,j.
• Multiplication to all elements of any one row (ith) by a fixed non-zero scalar (λ).
�denoted by Ri(λ).
• Addition to all elements of any one row (ith) a fixed scalar (λ) multiple of corresponding elements of another row (jth).
�denoted by Ri,j(λ).
UNIT-II
30
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
�denoted by Ci,j.
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
�denoted by Ci,j.
• Multiplication to all elements of any one column (ith) by a fixed non-zero scalar (λ).
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
�denoted by Ci,j.
• Multiplication to all elements of any one column (ith) by a fixed non-zero scalar (λ).
�denoted by Ci(λ).
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
�denoted by Ci,j.
• Multiplication to all elements of any one column (ith) by a fixed non-zero scalar (λ).
�denoted by Ci(λ).
• Addition to all elements of any column (ith) a fixed scalar (λ) multiple of corresponding elements of another column (jth).
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Column Operations • There are three types of elementary column
operations or transformations :
• Interchange of any two (ith and jth) column.
�denoted by Ci,j.
• Multiplication to all elements of any one column (ith) by a fixed non-zero scalar (λ).
�denoted by Ci(λ).
• Addition to all elements of any column (ith) a fixed scalar (λ) multiple of corresponding elements of another column (jth).
�denoted by Ci,j(λ).
UNIT-II
31
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Operations
UNIT-II
32
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Operations • The three elementary row operations and the
three elementary column operations together constitute 6 elementary operations.
UNIT-II
32
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Operations • The three elementary row operations and the
three elementary column operations together constitute 6 elementary operations.
• The elementary operations alter the matrix.
UNIT-II
32
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Operations • The three elementary row operations and the
three elementary column operations together constitute 6 elementary operations.
• The elementary operations alter the matrix.
• The elementary operations do not alter order of a matrix.
UNIT-II
32
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Elementary Operations • The three elementary row operations and the
three elementary column operations together constitute 6 elementary operations.
• The elementary operations alter the matrix.
• The elementary operations do not alter order of a matrix.
• The elementary operations do not alter rank of a matrix.
UNIT-II
32
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
• Ri(λ)−1 = Ri(1/λ)
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
• Ri(λ)−1 = Ri(1/λ)
• Ri,j(λ)−1 = Ri,j(−λ)
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
• Ri(λ)−1 = Ri(1/λ)
• Ri,j(λ)−1 = Ri,j(−λ)
• Ci,j−1 = Ci,j
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
• Ri(λ)−1 = Ri(1/λ)
• Ri,j(λ)−1 = Ri,j(−λ)
• Ci,j−1 = Ci,j
• Ci(λ)−1 = Ci(1/λ)
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse of Elementary Operations • Inverse of elementary operations are
elementary operations of same type.
• Ri,j−1 = Ri,j
• Ri(λ)−1 = Ri(1/λ)
• Ri,j(λ)−1 = Ri,j(−λ)
• Ci,j−1 = Ci,j
• Ci(λ)−1 = Ci(1/λ)
• Ci,j(λ)−1 = Ci,j(−λ)
UNIT-II
33
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row & Column Equivalent Matrices
UNIT-II
34
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row & Column Equivalent Matrices • Two matrices are said to be row-equivalent
⇔ one can be obtained from the other by a succession of finite number of elementary row operations.
UNIT-II
34
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row & Column Equivalent Matrices • Two matrices are said to be row-equivalent
⇔ one can be obtained from the other by a succession of finite number of elementary row operations.
• Two matrices are said to be column-
equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary column operations.
UNIT-II
34
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row & Column Equivalent Matrices • Two matrices are said to be row-equivalent
⇔ one can be obtained from the other by a succession of finite number of elementary row operations.
• Two matrices are said to be column-
equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary column operations.
• Two matrices are said to be equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary operations.
UNIT-II
34
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix • A matrix is said to be in row-echelon form
⇔
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix • A matrix is said to be in row-echelon form
⇔
�Non-zero rows are on the top and zero rows are in the bottom
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix • A matrix is said to be in row-echelon form
⇔
�Non-zero rows are on the top and zero rows are in the bottom
�First non-zero entry (leading entry) in each non-zero row is 1.
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix • A matrix is said to be in row-echelon form
⇔
�Non-zero rows are on the top and zero rows are in the bottom
�First non-zero entry (leading entry) in each non-zero row is 1.
�The position of the leading entry is strictly increasing with rows.
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Echelon Matrix • A matrix is said to be in row-echelon form
⇔
�Non-zero rows are on the top and zero rows are in the bottom
�First non-zero entry (leading entry) in each non-zero row is 1.
�The position of the leading entry is strictly increasing with rows.
• A row-reduced echelon matrix is a row echelon matrix in which all other elements of a column containing leading entry, except that leading entry, are 0.
UNIT-II
35
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix • A matrix is said to be in column-echelon
form ⇔
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix • A matrix is said to be in column-echelon
form ⇔
�Non-zero columns are on the left and zero columns are in the right
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix • A matrix is said to be in column-echelon
form ⇔
�Non-zero columns are on the left and zero columns are in the right
�First non-zero entry (leading entry) in each non-zero column is 1.
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix • A matrix is said to be in column-echelon
form ⇔
�Non-zero columns are on the left and zero columns are in the right
�First non-zero entry (leading entry) in each non-zero column is 1.
�The position of the leading entry is strictly increasing with columns.
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Column Echelon Matrix • A matrix is said to be in column-echelon
form ⇔
�Non-zero columns are on the left and zero columns are in the right
�First non-zero entry (leading entry) in each non-zero column is 1.
�The position of the leading entry is strictly increasing with columns.
• A column-reduced echelon matrix is a column echelon matrix in which all other elements of a row containing leading entry, except that leading entry, are 0.
UNIT-II
36
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix • Number of linearly independent rows in a
matrix is its row rank.
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix • Number of linearly independent rows in a
matrix is its row rank.
• Number of linearly independent columns in a matrix is its column rank.
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix • Number of linearly independent rows in a
matrix is its row rank.
• Number of linearly independent columns in a matrix is its column rank.
• Row rank of a matrix
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix • Number of linearly independent rows in a
matrix is its row rank.
• Number of linearly independent columns in a matrix is its column rank.
• Row rank of a matrix
= column rank of the matrix
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Row Rank & Column Rank of Matrix • Number of linearly independent rows in a
matrix is its row rank.
• Number of linearly independent columns in a matrix is its column rank.
• Row rank of a matrix
= column rank of the matrix
= rank of the matrix.
UNIT-II
37
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Working Rule for Finding Rank
UNIT-II
38
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Working Rule for Finding Rank • Working procedure for finding the row rank of
a matrix is that the matrix is reduced to row echelon form by applications of elementary row operations and then the number of non-zero rows in the matrix are counted which is the row rank.
UNIT-II
38
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Working Rule for Finding Rank • Working procedure for finding the row rank of
a matrix is that the matrix is reduced to row echelon form by applications of elementary row operations and then the number of non-zero rows in the matrix are counted which is the row rank.
• Working procedure for finding the column rank of a matrix is that the matrix is reduced to column echelon form by applications of elementary column operations and then the number of non-zero columns in the matrix are counted which is the column rank.
UNIT-II
38
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations
UNIT-II
39
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • A system of m simultaneous linear equations
in n unknowns is homogeneous ⇔ pure non-coefficient constants are zero, else it is non-homogeneous.
UNIT-II
39
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • A system of m simultaneous linear equations
in n unknowns is homogeneous ⇔ pure non-coefficient constants are zero, else it is non-homogeneous.
• A system of m simultaneous linear equations in n unknowns is consistent ⇔ it has a common solution, else it is inconsistent.
UNIT-II
39
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • A system of m simultaneous linear equations
in n unknowns is homogeneous ⇔ pure non-coefficient constants are zero, else it is non-homogeneous.
• A system of m simultaneous linear equations in n unknowns is consistent ⇔ it has a common solution, else it is inconsistent.
• Homogeneous system always has a trivial solution in which all unknowns are zero.
UNIT-II
39
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • A system of m simultaneous linear equations
in n unknowns is homogeneous ⇔ pure non-coefficient constants are zero, else it is non-homogeneous.
• A system of m simultaneous linear equations in n unknowns is consistent ⇔ it has a common solution, else it is inconsistent.
• Homogeneous system always has a trivial solution in which all unknowns are zero.
• Homogeneous system of n equations in n unknowns has non-trivial solution ⇔ determinant of coefficients is zero.
UNIT-II
39
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations
UNIT-II
40
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • Non-homogeneous system of n equations in n
unknowns has a solution ⇔ determinant of coefficients is non-zero.
UNIT-II
40
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • Non-homogeneous system of n equations in n
unknowns has a solution ⇔ determinant of coefficients is non-zero.
• General system AX = B has a solution ⇔ ρ(A) = ρ([A|B])
UNIT-II
40
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • Non-homogeneous system of n equations in n
unknowns has a solution ⇔ determinant of coefficients is non-zero.
• General system AX = B has a solution ⇔ ρ(A) = ρ([A|B])
• For consistent system ρ(A) < n, there are infinitely many solutions with n − r parameters.
UNIT-II
40
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
System of Linear Equations • Non-homogeneous system of n equations in n
unknowns has a solution ⇔ determinant of coefficients is non-zero.
• General system AX = B has a solution ⇔ ρ(A) = ρ([A|B])
• For consistent system ρ(A) < n, there are infinitely many solutions with n − r parameters.
• For consistent system ρ(A) = n, there is unique solution.
UNIT-II
40
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
A − λI is called characteristic matrix of A.
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
A − λI is called characteristic matrix of A.
|A − λI| is called characteristic polynomial of A.
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
A − λI is called characteristic matrix of A.
|A − λI| is called characteristic polynomial of A.
|A − λI| = 0 is called characteristic equation of A.
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
A − λI is called characteristic matrix of A.
|A − λI| is called characteristic polynomial of A.
|A − λI| = 0 is called characteristic equation of A.
The roots of characteristic equation are called as characteristic roots or eigenvalue or latent value or proper value.
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • For a square matrix A :
A − λI is called characteristic matrix of A.
|A − λI| is called characteristic polynomial of A.
|A − λI| = 0 is called characteristic equation of A.
The roots of characteristic equation are called as characteristic roots or eigenvalue or latent value or proper value.
Spectrum of A is the collection of all characteristic roots of A.
UNIT-II
41
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Vectors
UNIT-II
42
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Vectors Characteristic vector of a square matrix A for characteristic value λ is the vector X satisfying relation AX = λX.
UNIT-II
42
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Vectors Characteristic vector of a square matrix A for characteristic value λ is the vector X satisfying relation AX = λX.
• One characteristic vector cannot correspond to more than one characteristic values.
UNIT-II
42
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Vectors Characteristic vector of a square matrix A for characteristic value λ is the vector X satisfying relation AX = λX.
• One characteristic vector cannot correspond to more than one characteristic values.
• One characteristic value corresponds to infinitely many characteristic vectors.
UNIT-II
42
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Vectors Characteristic vector of a square matrix A for characteristic value λ is the vector X satisfying relation AX = λX.
• One characteristic vector cannot correspond to more than one characteristic values.
• One characteristic value corresponds to infinitely many characteristic vectors.
• Cayley-Hamilton Theorem : Every square matrix satisfies its own characteristic equation.
UNIT-II
42
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities
UNIT-III
43
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities • Complex number is of form x + iy, where x
and y are real numbers and i = .
UNIT-III
43
1−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities • Complex number is of form x + iy, where x
and y are real numbers and i = .
• Complex number can be written using trigonometric functions as x + iy = r(cosθ + isinθ)
UNIT-III
43
1−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities • Complex number is of form x + iy, where x
and y are real numbers and i = .
• Complex number can be written using trigonometric functions as x + iy = r(cosθ + isinθ)
• x = real part, y = imaginary part
UNIT-III
43
1−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities • Complex number is of form x + iy, where x
and y are real numbers and i = .
• Complex number can be written using trigonometric functions as x + iy = r(cosθ + isinθ)
• x = real part, y = imaginary part
• r = modulus , θ = argument/amplitude
UNIT-III
43
1−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Trigonometry : Complex Quantities • Complex number is of form x + iy, where x
and y are real numbers and i = .
• Complex number can be written using trigonometric functions as x + iy = r(cosθ + isinθ)
• x = real part, y = imaginary part
• r = modulus , θ = argument/amplitude
• DeMoivre’s Thoerem : (cosθ + isinθ)n = (cosnθ + isinnθ)
UNIT-III
43
1−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of cosnθ and sinnθ
UNIT-III
44
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of cosnθ and sinnθ
UNIT-III
44
2 2( 1)cos cos cos sin
2!n nn n
nθ θ θ θθ θ θ θθ θ θ θθ θ θ θ−−−−−−−−= − ±= − ±= − ±= − ±⋯⋯⋯⋯
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of cosnθ and sinnθ
UNIT-III
44
2 2( 1)cos cos cos sin
2!n nn n
nθ θ θ θθ θ θ θθ θ θ θθ θ θ θ−−−−−−−−= − ±= − ±= − ±= − ±⋯⋯⋯⋯
1
3 3
sin cos sin
( 1)( 2)cos sin
3!
n
n
n n
n n n
θ θ θθ θ θθ θ θθ θ θ
θ θθ θθ θθ θ
−−−−
−−−−
====− −− −− −− −− ±− ±− ±− ±⋯⋯⋯⋯
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of cos and sin
UNIT-III
45
2 4 6
cos 12! 4! 6!
α α αα α αα α αα α ααααα = − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of cos and sin
UNIT-III
45
2 4 6
cos 12! 4! 6!
α α αα α αα α αα α ααααα = − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯
3 5 7
sin3! 5! 7!
α α αα α αα α αα α αα αα αα αα α= − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of Powers of cos and sin
UNIT-III
46
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of Powers of cos and sin
UNIT-III
46
12 cos cos cos( 2)
( 1)cos( 4)
2!
n n n n n
n nn
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of Powers of cos and sin
UNIT-III
46
12 cos cos cos( 2)
( 1)cos( 4)
2!
n n n n n
n nn
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯
1 22 ( 1) sin cos cos( 2)
( 1)cos( 4) for even
2!
nn n n n n
n nn n
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−− − = − −− = − −− = − −− = − −−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Expansions of Powers of cos and sin
UNIT-III
46
12 cos cos cos( 2)
( 1)cos( 4)
2!
n n n n n
n nn
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯
1 22 ( 1) sin cos cos( 2)
( 1)cos( 4) for even
2!
nn n n n n
n nn n
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−− − = − −− = − −− = − −− = − −−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯
11 22 ( 1) sin sin sin( 2)
( 1)sin( 4) for odd
2!
nn n n n n
n nn n
θ θ θθ θ θθ θ θθ θ θ
θθθθ
−−−−−−−− − = − −− = − −− = − −− = − −
−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Exponential Series, circular functions
UNIT-III
47
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Exponential Series, circular functions
UNIT-III
47
2 3
12! 3!
x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Exponential Series, circular functions
UNIT-III
47
2 3
12! 3!
x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯
2 31 11 1 1
2! 3! 2! 3!
xx x
x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +
⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Exponential Series, circular functions
UNIT-III
47
2 3
12! 3!
x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯
2 31 11 1 1
2! 3! 2! 3!
xx x
x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +
⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯
sin2
ix ixe ex
i
−−−−−−−−====
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Exponential Series, circular functions
UNIT-III
47
2 3
12! 3!
x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯
2 31 11 1 1
2! 3! 2! 3!
xx x
x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +
⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯
sin2
ix ixe ex
i
−−−−−−−−====
cos2
ix ixe ex
−−−−++++====
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E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
2sech
x xx
e e−−−−====++++
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
2sech
x xx
e e−−−−====++++
cothx x
x x
e ex
e e
−−−−
−−−−
++++====−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
• cos(ix) = coshx
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
2sech
x xx
e e−−−−====++++
cothx x
x x
e ex
e e
−−−−
−−−−
++++====−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
• cos(ix) = coshx
• sin(ix) = isinhx
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
2sech
x xx
e e−−−−====++++
cothx x
x x
e ex
e e
−−−−
−−−−
++++====−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Hyperbolic Functions
• cos(ix) = coshx
• sin(ix) = isinhx
• tan(ix) = itanhx
UNIT-III
48
sinh2
x xe ex
−−−−−−−−====
cosh2
x xe ex
−−−−++++====
tanhx x
x x
e ex
e e
−−−−
−−−−
−−−−====++++
2cosech
x xx
e e−−−−====−−−−
2sech
x xx
e e−−−−====++++
cothx x
x x
e ex
e e
−−−−
−−−−
++++====−−−−
Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra and Trigonometry ALGEBRA AND TRIGONOMETRY
49