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MATRICES
Class 10 Notes
Introduction: “MATRIX’ is a Latin word for
womb. The knowledge of matrices is
necessary in various branches of
mathematics. Matrices are one of the most
powerful tools in mathematics. The
evolution of concept of matrices is the
result of an attempt to obtain compact and
simple methods of solving system of linear
equations. Matrix notation and operations
are used in electronic spread sheet
programs in PC which can be used in
different areas of business and science.
Matrices are also used in cryptography.
Matrix: A rectangular arrangement of set
of elements in the form of horizontal and
vertical lines is called matrix. The
elements can be numbers or variables.
Example:
Numbers arranged in the form of
horizontal lines are called rows. Numbers
arranged in the form of vertical lines are
called columns. Elements of a matrix are
represented in pair of brackets (or)
parenthesis.
i.e., [ ] (or) ( ). Matrices is the plural of
matrix.
Example:
Matrix A contains 2 rows and 3
columns.
Order of a matrix: A matrix having ‘m’
rows and ‘n’ columns is said to be of
order m ⋅ n (m cross n (or) m by n).
If matrix A has ‘m’ rows and ‘n’
columns then m ⋅ n (m cross n (or) m
by n) is the order of matrix, and is
denoted by A m ⋅n.
Example:
Number of rows = 2
Number of columns = 2
∴ Order of matrix = 2 ⋅ 2.
It is denoted by A 2 ⋅ 2
Example: The order of the matrix
is:
Solution: Number of rows = 2.
Number of columns = 3.
∴ Order of matrix B = 2 ⋅ 3.
Generally, we use capital letters to denote
matrices.
It is denote by B 2 ⋅ 3
Example: The order of matrix is:
Solution: Number of rows = 1.
Number of columns = 4.
∴ Order of matrix C = 1 ⋅ 4.
It is denote by C 1 ⋅ 4
Example: The order of matrix D = [ 1 ] is
Solution: Number of rows = 1 = Number of columns.
∴ Order of matrix D = 1 ⋅ 1.
It is denote by D 1 ⋅ 1
A matrix having m rows and n columns having mn (or) nm number of elements.
Example:
Order of matrix = 2 ⋅ 3 The element 6 occurs in first row and third column. Therefore it can be represented as
a 13 = 6.
General form of a matrix:
In general, m ⋅ n matrix can be represented as
where
a ij is the element of the matrix in i th row and j th column i.e., is called element
of the matrix.
Example: Construct a 2 ⋅ 3 matrix whose elements are defined by for the representation of elements.
Solution: Given
i = 1, 2; j = 1, 2, 3
∴ Required matrix =
Example: Construct a 3 ⋅ 2 matrix whose elements are defined by
Solution:
i = 1, 2, 3; j = 1, 2
∴ Required matrix
Example: Construct a 2 ⋅ 2 matrix if for the representation of elements.
Solution: Given i = 1, 2; j = 1, 2
∴ Required matrix = Types of matrices:
i) Row matrix: A matrix having only one row is called row matrix.
Example:
Order of any row matrix is , where n is number of column, n = 2, 3, 4….
ii) Column matrix: A matrix having only one column is called column
matrix.
Example:
Order of any column matrix is , where m is number of column, m = 2, 3, 4….
iii) Rectangular matrix: A matrix in which number of rows are not equal to
number of columns is called rectangular matrix.
Example:
iv) Square matrix: A matrix having equal number of rows and columns is
called a square matrix.
is a square matrix if m = n. A matrix of order is called a square
matrix of order m.
Example:
If is a square matrix of order n, then the elements a 11 , a 22, …. a nn constitute
principal diagonal. Hence a ij is an element of the diagonal if i = j (or) non-diagonal if i
≠ j.
Example:
, a, e, k are the elements of the principal diagonal.
The sum of elements of the diagonal of a square matrix. A is called trace of A is denoted by
Tr(A).
Example: , find the trace of A?
Solution: Elements of principal diagonal are 1, 0, 9.
Tr(A) = 1 + 0 + 9 = 10.
v) Diagonal matrix: If each non-diagonal element of a square matrix is
equal to zero, then the matrix is called a diagonal matrix.
If is a diagonal matrix, then . It is denoted as
.
Example:
Example: Determine the diagonal elements of the matrix
Solution: i = j
a11 = a, a22 = B, a 33 = C1
∴ Diagonal elements of the matrix are a, B, C 1 .
Example: Determine the principal diagonal of the matrix given by
of order 3 ⋅ 3
Solution: Given
vi) Scalar matrix: If each non-diagonal element of a square matrix is zero
and all diagonal elements are same, then it is called a scalar matrix.
A matrix is said to be a scalar matrix if.
Example:
vii) Unit matrix (or) Identity matrix: If each non-diagonal element of a
square matrix is zero and each diagonal element is equal to 1, then that matrix is
called a unit or identity matrix. It is denoted by I n (or) I.
If
Example:
Example:
viii) Zero (or) Null matrix: If each element of a matrix is zero, then it is called
a null matrix or zero matrix. It is denoted by O m ⋅n (or) O.
Example:
A square matrix is said to be upper triangular if a ij = 0 for all i > j .
A square matrix is said to be lower triangular if aij = 0 for all i < j . Example:
i) upper triangular matrices.
ii) lower triangular matrices.
Comparison of matrices: Two matrices can be compared if the order of matrices is
equal. i.e., they have same number of rows and same number of columns.
Example:
Compared because order of matrices is equal.
Equality of matrices: Matrices A and B are said to be equal if A and B are of same
order and the corresponding elements of A and B are equal.
Thus, A = B if aij = bij .
Example:
Example: If such that A = B then find x, y, z .
Solution:
Given A = B, corresponding elements are equal.
Multiplication of a matrix by a scalar:
If every element of a matrix A is multiplied by a number (real or complex) k, the
matrix obtained is k times A and is denoted by kA and the operation is called scalar
multiplication.
Example:
1.
Solution:
1. If a and b are any two scalars and P is a matrix, then a(bP) = (ab)P
2. If m and n are any two scalars and A is a matrix, then (m + n)A = mA +
nA.
Addition of matrices: Two matrices can be added if the order of matrices are equal.
The sum matrix of two matrices A and B is obtained by adding the corresponding
elements of A and B.
If A = and B = then
A + B can be represented as such that i = 1, 2, 3, ….. m, j =1, 2, 3, ….. n.
Example:
Example: Find the sum matrix of matrices and
Solution: The given two matrices are of same order. So they can be added.
Sum =
Example: Find the sum of matrices A = aij = 2i + j and B = bij = j
i .
Solution: A = [ a ij ]
Given a ij = 2i + j
i = 1, 2; j = 1, 2
B = [ bij ]
Given aij = ji
i = 1, 2; j = 1, 2
Properties on addition of matrices:
i) Closure property: When we add any two matrices of same order then the
resultant also is matrix of same order.
Example:
ii) Commutative property: Let A, B be two matrices then A + B = B + A
Example:
iii) Associative property: Let A, B, C be three matrices then (A + B) + C = A +
(B + C)
Example:
iv) Additive identity: Let A be a matrix and O is zero matrix then A + O = O + A = A
Example:
v) Additive inverse: Let A be a matrix then A + (-A) = (-A) + A = O
∴ (-A) is called additive inverse of A.
A is called additive inverse of –A.
Example:
vi) Cancellation laws:
A + B = A + C ⇒ B = C (Left cancellation law)
B + A = C + A ⇒ B = C (Right cancellation law)
Subtraction of matrices: Two matrices can be subtracted if they are of the same order.
The difference of two matrices of same order A and B i.e. A – B is obtained by
subtracting the corresponding elements of B from that of A. Also the difference matrix
is of the same order as that of A or B.
If then A – B can be represented as such that
i = 1, 2, 3, ……m; j =1, 2, 3, ……n.
Example:
Example: Find the subtraction of matrices
Solution:
i)
Matrix subtraction is not commutative.
ii)
Matrix subtraction is not associative. Example: Find the subtraction of matrices if
Solution: Given
Aliter:
Example: Find A + (-B) if
Solution: Given
A + (-B) = A - B Solving a matrix equation: Let A, B be two matrices of same order such that A + X = B
where X is an unknown matrix or order equal to A and B matrices.
Example: find X of order 2 ⋅ 2 such that
i) X = A – B ii) A + X = B
Solution: Given
i) X = A – B
ii) A + X = B
Transpose of a matrix: The matrix obtained from any given matrix A by interchanging
its rows and columns is called transpose of the given matrix. It is denoted by A T .
If then A T =
Example:
Solution:
Example: If then find matrix.
Solution:
Properties of transpose matrix:
i) If A is any matrix then
ii) If A, B are two matrices of same order then
iii) If A, B are two matrices of same order then
iv) If A is a matrix and k is a scalar then
Example: If then find
i) ii) iii)
Solution: i) Given
ii)
iii)
Example: If and k is a scalar then verify that
Solution: Given
Symmetric matrix: A square matrix is said to be symmetrix if the transpose of the
matrix is equal to itself.
A square matrix is said to be symmetric if its ( i, j) th element is same as its
(j, i) th element, i.e., aij = aji , for all i, j.
Example:
∴ A is symmetric matrix.
i) Symmetric matrix is always a square matrix.
ii) A necessary and sufficient condition for a matrix A to be symmetric is
that it is equal to its transpose matrix. i.e., .
iii) Diagonal matrices are always symmetric.
Skew-symmetric matrix: If a square matrix and the negative of its transpose matrix
are equal then it is called a skew-symmetric matrix.
A square matrix A = [ aij ] is said to be skew-symmetric if the (i, j) th
element of A is the negative of the (j, i) th element of A. i.e., aij = -aji , for all i, j .
Example:
∴ A is skew symmetric matrix.
A matrix which is both symmetric and skew symmetric is called a square null matrix.
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