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B. Tech. (I sem), 2020-
MATHEMATICS-I (KAS-103T)
MODULE 1(MATRICES)
S.No MATRICES PAGE NO.
1.1 Introduction 2
1.2 Types of Matrices 2
1.3 Basic Operations on Matrices 7
1.4 Transpose of a Matrix 9
1.5 Symmetric Matrix 10
1.6 Skew-Symmetric Matrix 10
1.7 Complex Matrices 11
1.8 Hermitian and Skew-Hermitian matrix 13
1.9 Unitary^ Matrix^13
1.10 Elementary Transformations (or Operations) 19
1.11 Inverse of a Matrix by E-Operations (Gauss-Jordan Method) 19
1.12 Rank of a^ Matrix^26
1.13 Nullity of a Matrix 27
1.14 Methods of Finding Rank of Matrix 28
1.15 Solution of System of Linear Equations 46
1.16 Linear Dependence and Independence of Vectors 60
1.17 Eigenvalues and Eigenvector 63
1. 1.
1.
Cayley-Hamilton Theorem Similarity Transformation
Diagonalization of a matrix
75 80
80
SYLLABUS: Types of Matrices: Symmetric, Skew-symmetric and Orthogonal Matrices;
Complex Matrices, Inverse and Rank of matrix using elementary transformations, Rank-Nullity
theorem; System of linear equations, Characteristic equation, Cayley-Hamilton Theorem and its
application, Eigen values and Eigen vectors; Diagonalisation of a Matrix.
CONTENTS
MATRICES
1.1 INTRODUCTION
The term ‘Matrix’ was given by J.J.Sylvester about1850, but was introduced first by Cayley in
- By a ‘matrix’ we mean “rectangular array” of numbers. Matrices (plural of matrix) find
applications in solution of system of linear equations, probability, mathematical economics,
quantum mechanics, electrical networks, curve fitting, transportation problems etc. Matrices are
easily agreeable for computers.
Matrix inverse can be used in Cryptography. It can provide a simple and effective procedure for
encoding and decoding messages.
Matrix
A rectangular array of m. n numbers (real or complex) arranged in m rows (horizontal lines) and
n columns (vertical lines) and enclosed in brackets [ ] is called a matrix of order m n .It is
also called m n matrix.
The numbers in the matrix are called entries or elements of the matrix.
Elements of a matrix are located by the double subscript ij where i denotes the row and j the
column. The matrix A is written as
m m mn mn
n
n
a a a
a a a
a a a
A
...
... ... ... ...
...
...
1 2
21 22 2
11 12 1
If all these entries are real, then the matrix A is called a real matrix.
Note: Matrix has no numerical value.
1.2 TYPES OF MATRICES
(a) Square Matrix
In a matrix, if the number of rows = number of columns = n, then it is called a square matrix of
order n.
Example :
A is a square matrix since its order is 3 3.
Example:
Let
A
is a diagonal matrix of order 3.
(e) Scalar Matrix
In a diagonal matrix if all the diagonal elements are equal to a non-zero scalar , then it is called
a scalar matrix.
Example:
Let
A
is a scalar matrix of order 3.
(f) Unit Matrix or Identity matrix
In a diagonal matrix if all the diagonal elements are equal to 1, then it is called a unit matrix or
identity matrix.
Examples :
are identity matrices of orders 1, 2, 3 respectively.They are denoted by I 1 , I 2 , I 3.
In general , In is the identity matrix of order n.
(g) Zero Matrix or Null matrix
In a matrix (rectangular or square), if all the entries are equal to zero, then it is called a zero
matrix or null matrix.
Example:
A B
are zero matrices of type 3 ^3 , 2 (^4).
(h) Triangular Matrix
A square matrix A= [aij] is said to be upper triangular matrix if all the entries below the main
diagonal are zero. That is aij 0 if ij.
A square matrix A= [aij] is said to be lower triangular matrix if all the entries above the main
diagonal are zero. That is aij 0 if ij.
Examples:
The matrices
A
are upper triangular matrices.
and
The matrices
^ A are lower triangular matrices.
(i) Singular and Non-singular Matrices
A square matrix A is said to be singular if A 0 and non-singular if A 0.
Example:
A
is a singular matrix since A ^0.
Example: Let
I
Operating R 23 or C 23 on I, we get the elementary matrix
.It is denoted by E 23.
1.3 Basic Operations on Matrices
(a) Addition and subtraction of Matrices
Two matrices are said to be conformable for addition and subtraction if they have the same
order. If A and B are two matrices of the same order , then their sum A+B is a matrix each
element of which is obtained by adding the corresponding elements of A and B.
Let A= [aij] and B=[bij] be two matrices of the same type m n .Then A+B=[cij] where
cij=aij+bij for all i , j and A B is of type m n.
Similarly, if A and B are two matrices of the same order , then their difference A-B is a
matrix whose element are obtained by subtracting the elements of B from the
corresponding elements of A.
Let A= [aij] and B = [bij] be two matrices of the same type m n .Then A-B = [cij] where
cij=aij-bij for all i , j
.
Example : If
^ A then
A B
We see that A^ and B are of type 2 ^3 and A B is also of the type^2 ^3
A B
(b) Scalar Multiplication of Matrices
Let A= [aij] be an m n matrix and k be a scalar. Then kA=[kaij].
Example:
If then^.
21 22 23
11 12 13
21 22 23
11 12 13
ka ka ka
ka ka ka
kA
a a a
a a a
A
Hence if k = -1, then.
21 22 23
11 12 13
a a a
a a a
A
( c) Multiplication of Matrices
If A and B are two matrices such that the number of columns of A is equal to the number of rows
of B, then the product AB is defined. Two such matrices are said to be conformable for
multiplication. In the product AB, A is known as pre-factor and B is known as post-factor.
Let Let A= [aij] be an m p matrix and B= [bij] be an p n matrix , then AB is defined and
AB=[cij] is an m n matrix, where.
1
p
k
cij aikbkj
That is cij is the sum of the products of the corresponding elements of the
th i row of A and the
th j column of B.
Example :
Let
andB
A
Since A is of type 3 3 and B is of the type 3 2 , AB is defined and AB is of type 3 (^2).
AB
Note: If A and B are square matrices of order n, then both AB and BA are defined,but not
necessarily equal. That is , AB ≠BA, in general.So, matrix multiplication is not commutative.
Example : If
,then
A A
Properties of Transpose of a Matrix
If A^ and B denote the transposes of A and B respectively, then
(a) A
A i.e., the transpose of the transpose of a matrix is the matrix itself.
(b) ^ A^ ^ B
A ^ B i.e., the transpose of the sum of two matrices is equal to the
sum of their transposes.
(c) ^ B^ ^ A
AB i.e., the transpose of the product of two matrices is equal to the
product of their transposes taken in the reverse order.
1.5: Symmetric Matrix
A square matrix A^ a^ ijis said to be symmetric if A Ai.e., if the transpose of the matrix
is equal to the matrix itself.
Thus, for a symmetric matrix A a^ ij, aij a ji.
Example :
g f c
h b f
a h g
are symmetric matrices.
1.6: Skew-Symmetric Matrix
A square matrix A a (^) ijis said to be symmetric if A Ai.e., if the transpose of the matrix
is equal to the negative of the matrix.
Thus, for a symmetric matrix A a (^) ij, aij a ji.
Putting j=i, a^ ii aii ^2 a^ ii ^0 or aii ^0 forall i. Thus, all diagonal elements of a skew-
symmetric matrix are zero.
Example :
g f
h f
h g
are skew-symmetric matrices.
Theorem: Every square matrix can uniquely be expressed as the sum of a
Symmetric matrix and a Skew-symmetric matrix.
Proof : Let A be any square matrix.
Consider B A A A A 2
andC 2
.
B C A A A A A 2
B ^ A A ^ A ^ A ^ ^ A A ^ ^ A A B
Bisasymmetricmatrix.
C A A A A A A A A C
Cisaskew-symmetricmatrix.
Hence every square matrix A can be expressed as A= B+C, B A A 2
is a symmetric matrix
and C A A
2
is a skew-symmetric matrix.
To prove that there is only one way in which A can be expressed as the sum of a symmetric
matrix and a skew-symmetric matrix, suppose
A=P+Q ……….. (1)
is another such representation in which P is a symmetric matrix, and Q is a skew-symmetric
matrix.
i.e. P ^ P and Q-Q
Then A P Q P Q P Q ...............( 2 )
From (1) and (2) , A A 2 P and A A 2 Q.
and Q 2
P A A B A A C so that P+Q =B+C
(ii) If A and B are two matrices conformable for product AB, then
.
AB B A
(iii) .
A B A B
(iv)^
kA kA where k is a scalar quantity.
1.8 Hermitian and Skew Hermition Matrix
A square matrix A is called Hermitian if A A.
In a Hermitian matrix , the diagonal elements are all real, while every other element is the
conjugate complex of the element in the transposed position.
Examples:
i i
i i
i i
A
i
i A are Hermitian matrix.
A square matrix A is called Skew- Hermitian if A A.
In a skew-Hermitian matrix , the diagonal elements are zero or purely imaginary numbers of
the form iβ where β is real. Every other element is the negative of the conjugate complex of the
element in the transposed position.
Examples :
i i
i i
i i
A
i i
i i A 3 4 5
are Skew-Hermitian matrix.
1.9 Unitary Matrix
A square matrix A is said to be unitary if AA I AA.
The determinant of a unitary matrix is of unit modulus. For a matrix to be unitary, it must be
non-singular.
Example :
i
i A is a Unitary matrix.
Question based on Hermitian, Skew Hermitian Matrix, Unitary matrix
Example1: If
i i
i i A 5 4 2
, verify that AA
is a Hermitian matrix where
A
is the conjugate transpose of A.
Solution:
i i
i
i
A
i i
i
i
A
A
i i
i i
i i
i
i
A A
(say)
B
i i
i i
i i
Now
i i
i i
i i
B
B
i i
i i
i i
B B
Hence B A A
(^) is a Hermitian matrix.
Example 2: If A and B are Hermitian, show that AB-BA is skew-Hermitian.
Solution: A and B are Hermitian A A B B
and
Now
AB BA AB BA
B A AB BA AB AB BA
AB BA is skew-Hermitian.
Example3: If A is a skew-Hermitian matrix, then show that iA is Hermitian.
Solution: A is a skew- Hermitian matrix A A
Now iA iA i A iA
A
i i
i i i
i i i
i i
i i
i i
Example 6: Show that the matrix
A
i
i is unitary.
Solution: Given
A
i
i
A
i
i A
Now. 0 1
I
i
i
i
i AA
Similarly AA I.
AA I AA.
Hence A is unitary.
Example 7: If
N
i
i is a matrix, then find
1 I-N I N and show that it
is a unitary matrix.
Solution: Given
N
i
i
Also
I
Now
i
i I-N
i
i I N
1 1 4 6 1 2 1
2
i i
i I N
Adj I N i
i
. 1 2 1
Thus I N
1
i
i AdjI N I N
. 1 2 1
Thus I-N I N
1
i
i
i
i
P,(say) 2 4 4
i
i
To prove P is unitary, we shall show PP I.
θ
We have ^
i
i P P
θ
i
i
i
i P P
θ
P P I
θ
Hence P i.e., (I-N)(I+N)
Practice Question:
1) Show that the matrix
i i
i i is unitary if^1.
2 2 2 2
2) If A is any square matrix, prove that A A AA A A
, , are all Hermitian and
A A^ is skew-Hermitian.
3) Show that
7 4i - 2 3 i
2 7 - 4i - 2 5i
A is a Hermitian matrix.
4) Show that
2 - i - 3 - 4i - 2i
3 2i 0 3 4i
i 3 2i - 2 i
A is a skew-Hermitian matrix.
5) If ,
5 3i -5i 2
2 - i 7 5 i
A
Show that A is a Hermitian matrix and iA is a skew-
Hermitian matrix.
1.10 Elementary Transformations (or Operations)
Any one of the following operations on a matrix is called an elementary transformation (or E-
operation).
(i) Interchange of two rows or two columns.
The interchange of i th and j th columns is denoted by Rij.
The interchange of i th and j th columns is denoted by Cij.
(ii) Multiplication of (each element of) a row or column by a non-zero number k.
The multiplication of i th row by k is denoted by Ri(k).
The multiplication of i th column by k is denoted by Ci(k).
(iii) Addition of k times the elements of a row (or column) to the corresponding
elements of another row(or column), k≠0.
The addition of k times the j th row to the i th row is denoted by Rij(k).
The addition of k times the j th column to the i th column is denoted by Cij(k).
If a matrix B is obtained from a matrix A by one or more E-operations, then B is said to
be equivalent to A. Two equivalent matrices A and B are written as A~B.
1.11 Inverse of a matrix by E-Operations (Gauss-Jordan Method)
The elementary row transformations which reduce a square matrix A to the unit matrix, when
applied to the unit matrix, gives the inverse matrix A
Consider only square matrices. Inverse of a n-square matrix A is denoted by A
- and is
defined such that
A A
- = A - A=I where I is n x n unit matrix.
To compute the inverse of a matrix, we use the concept of equivalent matrices.
(1) If we are to find out the inverse of a non-singular square matrix A, we first write A as
equivalent to I, a unit matrix of the same order.
A~I
(2) Then we apply elementary row operations on them. The objective is to reduce A to I. As
soon as this is achieved, the other matrix gives A
I~A
This is an elegant way of determining the inverse or reciprocal of a matrix A.
Properties of Inverse of a Matrix
(1) Inverse of A exists only if (^) A 0 i.e. A is non-singular matrix.
(2) Inverse of a matrix is unique.
(3) Inverse of a `product is the product of inverses in the reverse order
i.e., (AB)
(4) Transposition and inverse are commutative i.e.,(A
- ) T = (A T ) - , (A - ) - =A.
Example1: Find the inverse of A by Gauss-Jordan method where
A
Solution : Writing A=IA i.e.,
A
By R 21 (-2),R 31 (-3),we get
A
By R 23 A
By R 13 (3) ,R 23 (-3) A
By R 2 (-1) ,R 3 (-1) A
