1 Lecture May 2011 Matrices and Determinants Mathematics XI 2011.
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Transcript of 1 Lecture May 2011 Matrices and Determinants Mathematics XI 2011.
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Lecture May 2011
Matrices and Determinants
Mathematics XI 2011
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1.1 Matrices1.2 Operations of matrices1.3 Types of matrices1.4 Properties of matrices1.5 Determinants
1.61.7
Inverse of a 33 matrixApplication
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Zero matrices
Every element of a matrix is zero, it is called a zero matrix, i.e.,
0 0 0
0 0 0
0 0 0
A
1.1 Matrices
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Sums of matrices
1.2 Operations of matrices
Two matrices of the same order are said to be conformable for addition or subtraction.
Two matrices of different orders cannot be added or subtracted, e.g.,
are NOT conformable for addition or subtraction.
2 3 7
1 1 5
1 3 1
2 1 4
4 7 6
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1.3 Types of matrices
Identity matrix
The inverse of a matrix
The transpose of a matrix
Symmetric matrix
Orthogonal matrix
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(AB)-1 = B-1A-1
(AT)T = A and (A)T = AT
(A + B)T = AT + BT
(AB)T = BT AT
1.4 Properties of matrix
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1.5 Determinants
Consider a 2 2 matrix: 11 12
21 22
a aA
a a
Determinant of order 2
Determinant of A, denoted , is a number and can be evaluated by
11 1211 22 12 21
21 22
| |a a
A a a a aa a
| |A
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1.6 Inverse of a 33 matrix
Cofactor matrix of 1 2 3
0 4 5
1 0 6
A
The cofactor for each element of matrix A:
11
4 524
0 6A 12
0 55
1 6A 13
0 44
1 0A
21
2 312
0 6A 22
1 33
1 6A 23
1 22
1 0A
31
2 32
4 5A 32
1 35
0 5A 33
1 24
0 4A
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Cofactor matrix of is then given by:
1 2 3
0 4 5
1 0 6
A
24 5 4
12 3 2
2 5 4
1.6 Inverse of a 33 matrix
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1.6 Inverse of a 33 matrix
Inverse matrix of is given by:1 2 3
0 4 5
1 0 6
A
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24 5 4 24 12 21 1
12 3 2 5 3 522
2 5 4 4 2 4
T
AA
12 11 6 11 1 11
5 22 3 22 5 22
2 11 1 11 2 11
1.7 Applications Cement Steel Floor(sq.ft) Banglow 200 Yds 300 2500 2000 Banglow 300 Yds 500 3500 3000 Banglow 500 Yds 750 5000 5500
Khi Lhr Isl 450 375 520330 350 410 120 90 150
Total Cost
Khi Lhr Isl
Banglow 200 Yds 1200000 1167500 1481000 Banglow 300 Yds 1740000 1682500 2145000 Banglow 500 Yds 2647500 2526250 3265000
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Students can also create parallel examples in
Financial institutionIndustriesTime management
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By : Syed Shujaat Hussain
Special Thanks to My Master Trainer Mr.
Adnan Kiyani
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