MAT

CANTOR

The knowledge of matrices is necessary in various branches of mathematics. Matrice

are one of the most powerful tools in mathematics. This mathematical tool simplities

our work to a great extent when compared with other straight forward methods. The

evolution of concept of matrices is the result of an attempt to obtain compact and

simple methods of solving system of linear equations. Matrices are not only used

representation of the coefficients in system of linear equations, but utility of matrices

far exceeds that use. Matrix notation and operations are used in electronic spreadsheet

programs for personal computer, which in turn is used in different areas of business

and science like budgeting, sales projection, cost estimation, analysing the results of a

experiment etc. Also, many physical operations such as

reflection through a plane can be represented mathematically by matrices. Matrices

are also used in cryptography. This mathematical tool is not only used in certain branches

of sciences, but also in genetics, economics, sociology, modern psychology and industrial

The essence of Mathematies lies in its freedom

3.1 Introduction

as a

magnification, rotation and

management.

In this chapter, we shall find it interesting to become acquainted with the

fundamentals of matrix and matrix algebra.

3.2 Matrix

Suppose we wish to express the information that Radha has 15 notebooks. We may

express it as [15] with the understanding that the number inside [] is the number of
notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks

and 6 pens. We may express it as [15 6] with the understanding that first number

inside [ is the number of notebooks while the other one is the number of pens possessed

y Radha. Let us now suppose that we wish to express the information of possession57

MATRICES

of notebooks and pens by Radha and her two friends Fauzia and Simran which

is as follows:

Radha

has

15

notebooks

6 pens

2 pens,

and

has

10

Fauzia

notebooks

and

has

13

Simran

notebooks

and
pens.

NOw this could be arranged in the tabular form as follows:

Notebooks

Pens

15

Radha

10

Fauzia

13

Simran

and this can be expressed as

<First row

15

Second row

10

13

Third row

First

Second
Column

Column

or

Simran

Radha

Fauzia

13

10

15

Notebooks

Pens

which can be expressed as:

First row

13

10

15

Second row

Third

Second

First
Column

Column

Column

In the first arrangement the entries in the first column represent the number of

note books possessed by Radha, Fauzia and Simran, respectively and the entries in the

second column represent the number of pens possessed by Radha, Fauzia and Simran,8

MATHEMATICS

respectively. Similarly, in the second arrangement, the entries in the first row represent

the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The

entries in the second row represent the number of pens possessed by Radha. Fauzia

and Simran, respectively. An arrangement or display of the above kind is called

matrix. Formally, we define matrix as:

Definition 1A matrix is an ordered rectangular array of numbers or functions. The

numbers or functions are called the elements or the entries of the matrix.

We denote matrices by capital letters. The following are some examples of matrices:

2+i 3

[1+ x

C=

2
-1

tan x

sin x+ 2

A=0 V5B 3.5

COSX

In the above examples, 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. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has

rows and 3 columns.

3.2.1 Order of a matrix

A matrix having m rows and n columns is called a matrix of order m Xn or simply m xn

matrix (read as an m by n matrix). So referring to the above examples of matrices, we

have A as 3 x 2 matrix, B as 3 x 3 matrix and C as 2 x 3 matrix. We observe that A has

3 x2 6 elements, B and C have 9 and 6 elements, respectively.

In general, an m xn matrix has the following rectangular array:

а,

а,2

a3. a

aи

aa
а,»

a..a

mn

or A= [a]Is ism, Isj<n i,je N

Thus the h row consists of the elements a ap, a, a while the jh column

consists of the elements a, a a, C

12

in

In general a, is an element lying in the h row and jth column. We can also call

it as the (i, j)th element of A. The number of elements in an m x n matrix will be

equal to mn.

C4...9

MATRICES

Note In this chapter

We shall follow the notation, namely A-la to indicate thatA is a matrix

of order mxn

2 We shall consider only those matrices whose elements are real numbers or

functions taking real values

We can also represent any point (x. y) in a plane by a matrix (column or row) as

or Lx. y). For example point P(O, 1) as a matrix representation may be given as

P=

or 10 11

Observe that in this way we can also express the vertices of a closed rectilinear

figure in the form of a matrix. For example, consider a quadrilateral ABCD with vertices
A (, 0). B (3, 2). C (1. 3). D (-1, 2)

Now. quadrilateral ABCD in the matrix form, can be represented as

CD

13

X=

1-11

B32

Y=

or

3 22x4

02

D- 2

Thus, matrices can be used as representation of vertices of geometrical figures in

a plane.

Now, let us consider some examples.

Example 1 Consider the following information regarding the number of men and women

workers in three factories I. II and III

Women workers

Men workers
25

30

31

25

II

26

27

III

Represent the above information in the form of a 3 x 2 matrix. What does the entry

in the third row and second column represent?

60

MATHEMATIC

Solution The information is represented in the form of a 3x2 matrix as follows

30 25

31

A 25

27 26

The entry in the third row and second column represents the mumber of women

workers in factory II

Example 2 If a matrix has 8 elements, what are the possible orders it can have?

Solution We know that if a matrix is of order mXn, it has mn elements. Thus, to find

all possible orders of a matrix with 8 elements, we will find all ordered pairs of naturat

numbers, whose product is 8.

Thus, all possible ordered pairs are (1, 8), (8. 1). (4. 2). (2, 4)
Hence, possible orders are 1 x 8, 8 x1,4 x 2, 2 x 4

Example 3 Construct a 3 x 2 matrix whose elements are given by a, -3

Solution In general a 3 x 2 matrix is given by Aa

a2

=-3) ), 1 = 1,2,3 and j 1,2

Now

11-3x11-1

Therefore

12-3x21=2

= 13-3x11=0

a31

Hence the required matrix is given by A=

MATRICES

61

3.3 Types of Matrices

In this section, we shall discuss different types of matrices.

)Column matrix

A matrix is said to be a column matrix if it has only one column.

For example, A=-1

1/2

is a column matrix of order 4 x 1.

in general. A = |a 1
is a column matrix of order m x 1

imxl

(ii) Row matrix

A matrix is said to be a row matrix if it has only one row.

523

For example, B =

is a row matrix.

ix4

In general, B =

is a row matrix of order 1 x n

1 xn

Square matrix

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 x n matrix is said to be a square matrix if

m n and is known as a square matrix of order 'n'

-1

32

For example A =

is a square matrix of order 3.

-1
In general, A = [a,l,

is a square matrix of order m.

i mxm

Note If A = [a] is a square matrix of order n, then elements (entries) a, a

Г1

-3

-1

are said to constitute the diagonal, of the matrix A. Thus, if A=2

Then the elements of the diagonal of A are 1, 4, 6.

32

62

MATHEMATICS

(iv) Diagonal matrix

A square matrix B- is said to be a diaeoal mtri all s on

diagonal elements are rer, that is a matrix B- is said to be a diago

matrix if b0, wheni

[-1.1 0 0

T-1

20

For example. A = (41, B-|

of order 1. 2. 3, respectively

(v) Scalar matrix

that is, a square matrix B = [b is said to be a scalar matrix if

when i #j

b=0

= j. for some constant k

when i

b = k.

For example

30

A = [3].

C=

В-

are scalar matrices of order 1, 2 and 3, respectively

(vi) Identity matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero

is called an identity matrix. In other words, the square matrix A = [a] is an

1 if i-j

O if i#j

identity matrix, if a

We denote the identity matrix of order n by I. When order is clear from the

context, we simply write it as I.

10

For example [I],

are identity matrices of order 1, 2 and 3,

respectively.

Observe that a scalar matrix is an identity matrix when k

matrix is clearly a scalar matrix.

1. But every identity

MATRICES

63

(vii) Zero nmatrix

A matrix is said to be zero matrix or null matrix if all its elements are zero.

For example. [0]

(0, 01 are all zero matrices. We denote

zero matrix by O. Its order will be clear from the context.

3.3.1 Equality of matrices

Definition 2 Two matrices A = la,l and B = [b,] are said to be equal if

)they are of the same order

all i and j.

23

For example,

and

are equal matrices but

and

are

not equal matrices. Symbolically, if two matrices A and B are equal, we write A = B.

-1.5

then x1.5, y 0, z

2, a = 16, b 3, c 2

a2

If

bc
2

3 z+4 2y-7

6 3y-2

-3 2c 2

-6 a-1

=6

Example 4 If

b-3

2b+ 4 -21

-21

Find the values of a, b, c, x, y and z.

Solution As the given matrices are equal, therefore, their corresponding elements

must be equal. Comparing the corresponding elements, we get

2y 7 3y - 2

z+ 4=6,

x+3=0,

b - 3 2b + 4,

0 2c+ 2

a- 1=-3,

Simplifying, we get

a=-2, b =-7, c = - 1, x = - 3, y = -5, z

2a+b a-2b

4-3

11 24

5c-d

4c+3d

(C m=m

D) None of these

9. Which of the given values of x and y make the following pair of matrices equal

x+7

y-2

2-3

y7

(A)=

(B) Not possible to find

-1

D)x=

C)y7.

3
y

10. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is

(B) 18

(A) 27

(C) 81

D) 512

3.4 Operations on Matrices

In this section, we shall introduce certain operations on matrices, namely, addition of

matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices.

34.1 Addition of matrices

Suppose Fatima has two factories at places A and B. Each factory produces sport

shoes for boys and girls in three different price categories labelled 1, 2 and 3. The

quantities produced by each factory are represented as matrices given below:

Factory at A

Factory at B

Boys

Girls

Boys

Girls

80

60

90

50
2

75

65

70

55

90

85

75

75

Suppose Fatima wants to know the total production of sport shoes in each price

category. Then the total production

In category 1: for boys (80 + 90), for girls (60+ 50)

In category 2: for boys (75 +70), for girls (65+ 55)

In category 3: for boys (90+75), for girls (85+75)

80+90

60+50

65+55

75+70

This can be represented in the matrix form as

85+75

90+75

This new matrix is the sum of the above two malrices. We observe that the sum of
two matrices is a matrix obtained by adding the corresponding elements of the given

MATHEMATICS

66

matrices. Furthermore, the two matrices have to be of the same order

is another

is a 2 x 3 matrix and B

Thus, if A

2x3 matrix. Then, we define A +B=

In general, ifA = [a ] and B = [blare two matrices of the same order, say m x n.

Then, the sum of the two matrices A and B is defined as a matrixC= c, where

=a +b. for all possible values of i and j.

251

find A B

Example 6 Given A =

and B=

-2 3

Since A. B are of the same order 2 x 3. Therefore, addition of A and B is defined

and is given by

2+ 3 1+5 0

2+ 3 1+5 1-1

A+B =

1
0

2-2

3+3

0+

21

2P

Note

1. We emphasise that if A and B are not of the same order, then A + B is not

B=

12

defined. For example if A =

then A+ B is not defined.

2. We may observe that addition of matrices is an example of binary operation

on the set of matrices of the same order.

3.4.2 Multiplication of a matrix by a scalar

Now suppose that Fatima has doubled the production at a factory A in all categories

(refer to 3.4.1).

67

MATRICES
Previously quantities (in standard units) produced by factory A were

Boys

Girls

80

60

75

65

90

85

Revised quantities produced by factory A are as given below:

Boys Girls

I2x80 2x 60

2 2x75 2x 65

3 2x90 2x 85

160

120

130

We observe that

This can be represented in the matrix form as

150

180

170
the new matrix is obtained by multiplying each element of the previous matrix by 2

In general, we may define multiplication of a matrix by a scalar as follows: if

A = la]is a matrix and k is a scalar, then kA is another matrix which is obtained

by multiplying each element ofA by the scalar k.

In other words, kA = k [a] = [k (a)1that is, (i, jth element of kA is ka

for all possible values of i and j

mx

mx n

mxn

1 1.5

then

A =5 7 -3

For example, if

205

1 1.5 9

3 5 21

60

3 4.5

-9

3A=3 V5 7 -3

15

05

2
Negative of a matrix The negative of a matrix is denoted by - A. We define

-A (-1) A.

68

MATHEMATICS

A=

A is given by

then

For example, let

A (-)A (-1)

Difference of matrices If A = [a,l. B = |h, are two matrices of the same order

say mxn, then difference A - B is defined as a matrix D= |d, , where d a

for all value of i and j. In other words, D=A-B= A +(-1) B, that is sum of the matri

A and the matrix

-B.

then find 2A- B

3 -1

23

and B

Example 7 If A=

O2

3
2

Solution We have

123

-1

2A B = 2

02

3I

1 -3

-3

246

0-2

62

-1 5 3

2-3 4+1 6-3

6+0 2-2

4+1

3.4.3 Properties of matrix addition

The addition of matrices satisfy the following properties:

(i) Commutative Law If A = [a,], B = [b, are matrices of the same order, say

mxn, then A +B

B + A.
A+B la,] + [b,] = [a, + b,

Now

= [bi + a,l (addition of numbers is commutative)

= (b) + la,1) = B + A

(ii) Associative Law For any three matrices A = la], B

same order, saymx n, (A + B) + C = A + (B+C).

(A+B)+C = (la, + [b,) + [e

[b,l, C = [c,1 of the

Now

= [a, + (b, + c)l

= la,] + [(b, + c)1 = [a,] + (lb,] + Ic) = A + (B + C)

(Why?)

MATRICES

69

)Existence of additive identity Let A= lal be an m xn matrix and

a be an m x n zero matrix, then A + O 0+A = A. In other words. O is the

additive identity for matrix addition.

tiv) The existence of additive inverse Let A = la | be any matrix. then we

ave another matrix as - A =-a lsuch that A +(- A) = (- A) + A= O. So

-A is the additive inverse of A or ne gative of A.

3.4.4 Properties of scalar multiplication of a matrix

If A = la] and B = b| be two matrices of the same order, say m x n, and k and are

scalars, then

kA +B) k A+ kB. (ii) (k+ DA = k A + 1A

6i) k (A + B) = k (a]+ b,D

= Ik a] + Ik b,] = k la,] + k lb,1 = kA +kB

) (k+ D A = (k + D la]

= I(k + D a,l = Ik a ] + Va] = k la,] + I la,) = k A + 1 A

80

2 -2

Example 8 If A=4-2 and B

36

then find the matrix X, such that

-5 1

2A+3X= 5B.

Solution We have 2A +3X 5B

2A+3X-2A = 5B -2A

or

2A-2A+3X = 5B -2A

(Matrix addition is commutative)

or

(-2A is the additive inverse of 2A)

0+3X=5B- 2A

or

(O is the additive identity)

3X= 5B-2A

or
(5B -2A)

X=

or

10 -10

-16 0

-2]

80

-8

20 10 +

-2

-6 -12

-25 5

36

or

76

MATHEMATITCS

Note This does not mean that AB # BA for every pair of matrices A, Bf

which AB and BA. are defined. For instance,

1)

then AB= BA

B
If A=

Observe that multiplication of diagonal matrices of same order will be commutatiye

Zero matrix as the product of two non zero matrices

We know that, for real numbers a, b if ab 0, then either a = 0 or b= 0.ThiS nee

not be true for matrices, we will observe this through an example.

-1

and B=

Example 15 Find AB, if A -

35

Solution We have AB =

that

one of

Thus, if the product of two matrices is a zero matrix, it is not necessary

the matrices is a zero matrix.

3.4.6 Properties of multiplication of matrices

The multiplication of matrices possesses the following properties, which we state without

proof.

1. The associative law For any three matrices A, B and C. We have

A (BC), whenever both sides of the equality are defined.

(AB) C

2. The distributive law For three matrices A, B and C.

(i) A (B+C) AB + AC
AC + BC, whenever both sides of equality are defined.

(ii) (A+B) C

3. The existence of multiplicative identity For every square matrix A, there

exist an identity matrix of same order such that IA = AI = A.

Now, we shall verify these properties by examples.

-1

13

1 2 3-4

3, B=0 2 and C=

Example 16 If A2

find

2 0-2 1

A(BC), (AB)C and show that (AB )C

3 -1

-1 4

A(BC).

20. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen

physics books. 10 dozen economics books. Their selling prices are Rs 80, Rs 60

and Rs 40 each respectively. Find the total amount the bookshop will receive

from selling all the books using matrix algebra.

respectively Choose the correct answer in Exercises 21 and 22

21. The restriction on n, k and p so that PY +WY will be defined are:

(A) = 3. p=n

(C) P is arbitrary, k 3

If np, then the order of the matrix 7X- 5Z is:

(B) kis arbitrary, p 2

(D) k= 2, p 3

22.

(B) 2 x n

(A) PX2

(C) nx 3

(D) P X n

3.5. Transpose of a Matrix

In this section, we shall learn about transpose of a matrix and special types of matrices

such as symmetrie and skew symmetric matrices.

Definition 3 f A = [a ] be an m xn matrix, then the matrix obtained by interchanging

the rows and columns ofA is called the transpose of A. Transpose of the matrix A is

denoted by A' or (A). In other words, if A = [a]then A' = [a] For example,

nxm

330

if A =3

0-1
then A'=

-1

51

J2x3

3x2

3.5.1 Properties of transpose of the matrices

We now state the following properties of transpose of matrices without proof. These

be verified by taking suitable examples.

may

For any matrices A and B of suitable orders, we have

(i) (A) A,

( (AB) A+ B'

(ii) (kA) kA' (where k is any constant)

(iv) (A B) B' A

332

-1 21

and B =

Example 20 If A=

verify that

24

42

i) (A) A,
(kB) = kB', where k is any constant.

(ii) (A B) A B',

(iii)

4 B- 3 -6

A=

-2

-6

4 3 -6

AB

12

-24

then

15

30

A -2 4 51, B' =

Now

-2
4

3 [-2 4 5=6

B'A' =

15 (AB)

12

-6

12 -24

-30

(AB)' B'A'

Clearly

3,6 Symmetric and Skew Symmetric Matrices

Definition 4 A square matrix A = [a] is said to be symmetric if A'

la] = [a,] for all possible values of i and j.

A, that is,

-1.5 -1

is a symmetric matrix as A' = A

For example A =

-1

Definition 5 A square matrix A = [a] is said to be skew symmetric matrix if

A'- A, that is a-a for all possible values of i and j. Now, if we put i j, we
have a =- a Therefore 2a 0 or a, 0 for all i's.

This means that all the diagonal elements of a skew symmetric matrix are zero.

denoted by C,-C, +kC

For example, applying R,R,- 2R,, to C

we get

3.8 Invertible Matrices

Definition 6 If A is a square matrix of order m, and if there exists another square

matrix B of the same order m, such that AB = BA = 1, then B is called the inverse

matrix of A and it is denoted by A In that case A is said to be invertible.

A=

and B=

2 -3

For example, let

be two matrices.

-1

23

2 -3

AB

Now
2

4-3 -6+6

2-2 -3+4

BA =

=I. Thus B is the inverse of A, in other

Also

words B A and A is inverse of B, i.e., A B

Note

I. A rectangular matrix does not possess inverse matrix, since for products BA

and AB to be defined and to be equal, it is necessary that matrices A and B

should be square matrices of the same order.

2. If B is the inverse of A, then A is also the inverse of B.

Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.

Proof Let A = [a be a square matrix of order m. If possible, let B and C be two

inverses of A. We shall show that B C.

Since B is the inverse of A

AB BA I

(1)

Since C is also the inverse of A

(2

AC = CA = I

B BI B (AC) = (BA) C

IC = C
Thus

= BA-

Theorem 4 If A and B are invertible matrices of the same order, then (AB)