Chapter 3
Matrices
3.1 Overview
3.1.1 A matrix is an ordered rectangular array of numbers (or functions). For example,
x 4 3
A = 4 3 x
3 x 4
The numbers (or functions) are called the elements or the entries of the matrix.
The horizontal lines of elements are said to constitute rows of the matrix and the
vertical lines of elements are said to constitute columns of the matrix.
3.1.2 Order of a Matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply
m × n matrix (read as an m by n matrix).
In the above example, we have A as a matrix of order 3 × 3 i.e.,
3 × 3 matrix.
In general, an m × n matrix has the following rectangular array :
a11 a12 a13 a1n
a a22 a23 a2 n
21
A = [aij]m × n = 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N.
am1 am 2 am3 amn m × n
The element, aij is an element lying in the ith row and jth column and is known as the
(i, j)th element of A. The number of elements in an m × n matrix will be equal to mn.
3.1.3 Types of Matrices
(i) A matrix is said to be a row matrix if it has only one row.
, MATRICES 43
(ii) A matrix is said to be a column matrix if it has only one column.
(iii) A matrix in which the number of rows are equal to the number of columns,
is said to be a square matrix. Thus, an m × n matrix is said to be a square
matrix if m = n and is known as a square matrix of order ‘n’.
(iv) A square matrix B = [bij]n×n is said to be a diagonal matrix if its all non
diagonal elements are zero, that is a matrix B = [bij]n×n is said to be a
diagonal matrix if bij = 0, when i ≠ j.
(v) A diagonal matrix is said to be a scalar matrix if its diagonal elements are
equal, that is, a square matrix B = [bij]n×n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k.
(vi) A square matrix in which elements in the diagonal are all 1 and rest are
all zeroes is called an identity matrix.
In other words, the square matrix A = [aij]n×n is an identity matrix, if
aij = 1, when i = j and aij = 0, when i ≠ j.
(vii) A matrix is said to be zero matrix or null matrix if all its elements are
zeroes. We denote zero matrix by O.
(ix) Two matrices A = [aij] and B = [bij] are said to be equal if
(a) they are of the same order, and
(b) each element of A is equal to the corresponding element of B, that is,
aij = bij for all i and j.
3.1.4 Additon of Matrices
Two matrices can be added if they are of the same order.
3.1.5 Multiplication of Matrix by a Scalar
If A = [aij] m×n is a matrix and k is a scalar, then kA is another matrix which is obtained
by multiplying each element of A by a scalar k, i.e. kA = [kaij]m×n
3.1.6 Negative of a Matrix
The negative of a matrix A is denoted by –A. We define –A = (–1)A.
3.1.7 Multiplication of Matrices
The multiplication of two matrices A and B is defined if the number of columns of A is
equal to the number of rows of B.
, 44 MATHEMATICS
Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of
the matrices A and B is the matrix C of order m × p. To get the
(i, k)th element cik of the matrix C, we take the ith row of A and kth column of B,
multiply them elementwise and take the sum of all these products i.e.,
cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ain bnk
The matrix C = [cik]m×p is the product of A and B.
Notes:
1. If AB is defined, then BA need not be defined.
2. If A, B are, respectively m × n, k × l matrices, then both AB and BA are
defined if and only if n = k and l = m.
3. If AB and BA are both defined, it is not necessary that AB = BA.
4. If the product of two matrices is a zero matrix, it is not necessary that
one of the matrices is a zero matrix.
5. For three matrices A, B and C of the same order, if A = B, then
AC = BC, but converse is not true.
6. A. A = A2, A. A. A = A3, so on
3.1.8 Transpose of a Matrix
1. If A = [aij] be an m × n matrix, then the matrix obtained by interchanging
the rows and columns of A is called the transpose of A.
Transpose of the matrix A is denoted by A′ or (AT). In other words, if
A = [aij]m×n, then AT = [aji]n×m.
2. Properties of transpose of the matrices
For any matrices A and B of suitable orders, we have
(i) (AT)T = A,
(ii) (kA)T = kAT (where k is any constant)
(iii) (A + B)T = AT + BT
(iv) (AB)T = BT AT
3.1.9 Symmetric Matrix and Skew Symmetric Matrix
(i) A square matrix A = [aij] is said to be symmetric if AT = A, that is,
aij = aji for all possible values of i and j.
Matrices
3.1 Overview
3.1.1 A matrix is an ordered rectangular array of numbers (or functions). For example,
x 4 3
A = 4 3 x
3 x 4
The numbers (or functions) are called the elements or the entries of the matrix.
The horizontal lines of elements are said to constitute rows of the matrix and the
vertical lines of elements are said to constitute columns of the matrix.
3.1.2 Order of a Matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply
m × n matrix (read as an m by n matrix).
In the above example, we have A as a matrix of order 3 × 3 i.e.,
3 × 3 matrix.
In general, an m × n matrix has the following rectangular array :
a11 a12 a13 a1n
a a22 a23 a2 n
21
A = [aij]m × n = 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N.
am1 am 2 am3 amn m × n
The element, aij is an element lying in the ith row and jth column and is known as the
(i, j)th element of A. The number of elements in an m × n matrix will be equal to mn.
3.1.3 Types of Matrices
(i) A matrix is said to be a row matrix if it has only one row.
, MATRICES 43
(ii) A matrix is said to be a column matrix if it has only one column.
(iii) A matrix in which the number of rows are equal to the number of columns,
is said to be a square matrix. Thus, an m × n matrix is said to be a square
matrix if m = n and is known as a square matrix of order ‘n’.
(iv) A square matrix B = [bij]n×n is said to be a diagonal matrix if its all non
diagonal elements are zero, that is a matrix B = [bij]n×n is said to be a
diagonal matrix if bij = 0, when i ≠ j.
(v) A diagonal matrix is said to be a scalar matrix if its diagonal elements are
equal, that is, a square matrix B = [bij]n×n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k.
(vi) A square matrix in which elements in the diagonal are all 1 and rest are
all zeroes is called an identity matrix.
In other words, the square matrix A = [aij]n×n is an identity matrix, if
aij = 1, when i = j and aij = 0, when i ≠ j.
(vii) A matrix is said to be zero matrix or null matrix if all its elements are
zeroes. We denote zero matrix by O.
(ix) Two matrices A = [aij] and B = [bij] are said to be equal if
(a) they are of the same order, and
(b) each element of A is equal to the corresponding element of B, that is,
aij = bij for all i and j.
3.1.4 Additon of Matrices
Two matrices can be added if they are of the same order.
3.1.5 Multiplication of Matrix by a Scalar
If A = [aij] m×n is a matrix and k is a scalar, then kA is another matrix which is obtained
by multiplying each element of A by a scalar k, i.e. kA = [kaij]m×n
3.1.6 Negative of a Matrix
The negative of a matrix A is denoted by –A. We define –A = (–1)A.
3.1.7 Multiplication of Matrices
The multiplication of two matrices A and B is defined if the number of columns of A is
equal to the number of rows of B.
, 44 MATHEMATICS
Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of
the matrices A and B is the matrix C of order m × p. To get the
(i, k)th element cik of the matrix C, we take the ith row of A and kth column of B,
multiply them elementwise and take the sum of all these products i.e.,
cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ain bnk
The matrix C = [cik]m×p is the product of A and B.
Notes:
1. If AB is defined, then BA need not be defined.
2. If A, B are, respectively m × n, k × l matrices, then both AB and BA are
defined if and only if n = k and l = m.
3. If AB and BA are both defined, it is not necessary that AB = BA.
4. If the product of two matrices is a zero matrix, it is not necessary that
one of the matrices is a zero matrix.
5. For three matrices A, B and C of the same order, if A = B, then
AC = BC, but converse is not true.
6. A. A = A2, A. A. A = A3, so on
3.1.8 Transpose of a Matrix
1. If A = [aij] be an m × n matrix, then the matrix obtained by interchanging
the rows and columns of A is called the transpose of A.
Transpose of the matrix A is denoted by A′ or (AT). In other words, if
A = [aij]m×n, then AT = [aji]n×m.
2. Properties of transpose of the matrices
For any matrices A and B of suitable orders, we have
(i) (AT)T = A,
(ii) (kA)T = kAT (where k is any constant)
(iii) (A + B)T = AT + BT
(iv) (AB)T = BT AT
3.1.9 Symmetric Matrix and Skew Symmetric Matrix
(i) A square matrix A = [aij] is said to be symmetric if AT = A, that is,
aij = aji for all possible values of i and j.
