56 MATHEMATICS

v

The essence of Mathematics lies in its freedom. — CANTOR

v

3.1 Introduction

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

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

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 as a

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 an

experiment etc. Also, many physical operations such as magnification, rotation and

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

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

by Radha. Let us now suppose that we wish to express the information of possession

Chapter

3

MATRICES

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MATRICES 57

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

is as follows:

Radha has 15 notebooks and 6 pens,

Fauzia has 10 notebooks and 2 pens,

Simran has 13 notebooks and 5 pens.

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

Notebooks Pens

Radha 15 6

Fauzia 10 2

Simran 13 5

and this can be expressed as

or

Radha Fauzia Simran

Notebooks 15 10 13

Pens 6 2 5

which can be expressed as:

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,

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58 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 a

matrix. Formally, we define matrix as:

Definition 1

A

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:

5

– 2

A 0 5

3 6

=

,

1

2 3

2

B 3.5 –1 2

5

3 5

7

i

+ −

=

,

3

1 3

C

cos tan

sin 2

x

x

x x

x

+

=

+

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 2

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 × n or simply m × n

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

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

3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively.

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

or A = [a

ij

]

m × n

, 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N

Thus the i

th

row consists of the elements a

i1

, a

i2

, a

i3

,..., a

in

, while the j

th

column

consists of the elements a

1j

, a

2j

, a

3j

,..., a

mj

,

In general a

ij

, is an element lying in the i

th

row and j

th

column. We can also call

it as the (i, j)

th

element of A. The number of elements in an m × n matrix will be

equal to mn.

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A

Note In this chapter

1. We shall follow the notation, namely A = [a

ij

]

m ×

n

to indicate that A is a matrix

of order m × n.

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

x

y

(or [x, y]). For example point P(0, 1) as a matrix representation may be given as

0

P

1

=

or [0 1].

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 (1, 0), B (3, 2), C (1, 3), D (–1, 2).

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

2 4

A B C D

1 3 1 1

X

0 2 3 2

×

−

=

or

4 2

A 1 0

B 3 2

Y

C 1 3

D 1 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

Men workers Women workers

I 30 25

II 25 31

III 27 26

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

in the third row and second column represent?

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60 MATHEMATICS

Solution The information is represented in the form of a 3 × 2 matrix as follows:

30 25

A 25 31

27 26

=

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

workers in factory III.

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 m × n, it has mn elements. Thus, to find

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

numbers, whose product is 8.

Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)

Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4

Example 3 Construct a 3 × 2 matrix whose elements are given by

1

| 3 |

2

ij

a i j

= −

.

Solution In general a 3 × 2 matrix is given by

11 12

21 22

31 32

A

a a

a a

a a

=

.

Now

1

| 3 |

2

ij

a i j

= −

, i = 1, 2, 3 and j = 1, 2.

Therefore

11

1

|1 3 1| 1

2

a

= − × =

12

1 5

|1 3 2 |

2 2

a

= − × =

21

1 1

| 2 3 1|

2 2

a

= − × =

22

1

| 2 3 2 | 2

2

a

= − × =

31

1

| 3 3 1| 0

2

a

= − × =

32

1 3

| 3 3 2 |

2 2

a

= − × =

Hence the required matrix is given by

5

1

2

1

A 2

2

3

0

2

=

.

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3.3 Types of Matrices

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

(i) Column matrix

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

For example,

0

3

A 1

1/ 2

= −

is a column matrix of order 4 × 1.

In general, A = [a

ij

]

m × 1

is a column matrix of order m × 1.

(ii) Row matrix

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

For example,

1 4

1

B 5 2 3

2

×

= −