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7 ALGEBRA OF MATRICES

7.1 INTRODUCTION
Rajesh has two factories, one at Delhi and the other at Bombay. Each factory produces two items of garments for ladies and gents. The quantities produced by each factory is given in the matrices below: Factory at Delhi ITEM I Ladies Gents 600 300 ITEM II 550 450 Ladies Gents

Factory at Bombay
ITEM I 450 250 ITEM II 600 350

We are interested in finding out the total production of items. So what do we do? Or, we may be interested in finding the total cost of producing these items if cost per item is given for each type. In this lesson, we will be finding ways of answering such questions by going into addition, multiplication and algebra of matrices in general.

7.2 OBJECTIVES
After going through this lesson, you should be able to: state the condition for equality of two matrices multiply a matrix by a scalar find the sum of two matrices of the same order

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find the difference of two matrices of the same order state the condition for multiplication of two matrices multiply two matrices, if possible

7.3 PREVIOUS KNOWLEDGE
Concept of a matrix Order of a matrix Four fundamental operations and their properties Solution of problems using these properties

7.4 EQUALITY OF MATRICES
Consider the matrix 2 A –3 Its transpose will be A 2 –3 Observe that 1. 2. Order of matrix A = Order of matrix A', i.e., 2 × 2 Every element of A is same as the corresponding element of A'. In such a case, we say that Consider another example. 2 Let A = –1 3 –1 6 4 3 4 5 matrix A = matrix A'. –3 1 1 –3

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Its transpose will be 2 A' = –1 3 Again observe that (i) (ii) Order of matrix A = Order of matrix A' i.e., 3 × 3 Every element of A is same as the corresponding element of A'. ∴ we say that matrix A = matrix A' –1 6 4 3 4 5

Now consider the matrix A = 2 –1 3 4 6 7

Its transpose will be 2 A' = 3 6 –1 4 7

Are the two matrices equal ? We observe that (i) Order of A is 2×3 whereas order of A' is 3×2 hence order of A ≠ order of A' Every element of A is not equal to the corresponding element of A'. Therefore we can say that matrix A ≠ matrix A' Thus, we can define the equality of two matrices as:

(ii)

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Two matrices (i)

A and B are said to be equal if

they are of the same order and

(ii)

each element

of A is equal to the

corresponding element of B. For example: Consider two matrices of the same order x 5 and 2 5

When will the two matrices be equal? Matrices x = 5 5 2

if x = 2, since the two matrices are of the same order. Let us take some more examples Example A: Find the values of a and b if [a 3] = [4 b] Solution : If [a 3] = [4 b] then the corresponding elements of the matrices will be equal. ⇒ a = 4 and b = 3

Example B: Find the values of x and y if x 3 2 = –y 3 5 1 2

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Solution: If x 3 2 = –y 3 5 1 2

then their corresponding elements will be equal. ∴ x = 1 , y = –5

Example C: For what value of a, b, c and d will the two matrices a 6 –2 3 2b d and 1 6 –2 5c 4 2

be equal Solution :

3 5

Matrix

a 6

–2 3

2b = d

1 6

–2 5c

4 2

if the corresponding element of the two matrices will be equal. i.e. if

and if

and

{ {

a

= 1

2b = 4 5c = 3 d = 2 a = 1 b = 2 c = d =2

3 5

Thus, for a = 1, b = 2, c =

and d = 2, the matrix

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a 6 Example D:

–2 3

2b = d

1 6

–2 5c

4 2

For what value of a, b, c and d will the matrices a –3 a+c Solution: The matrix a –3 a+c b–2d 2b 7 = –3 4 6 7 5 1 b–2d 2b 7 = –3 4 6 7 5 1

If their corresponding elements are equal i.e. if

and if

and

{ {

a = 5 a+c = 4 b–2d = 1 2b = 6 a = 5 b = 3 c = –1 d = 1

Thus, for a = 5, b = 3, c = –1 and d = 1 the two given matrices will be equal. Check–point Tick the right choice for Q.1 and Q.2 1. Two matrices can be compared for equality if

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(i) (ii) 2.

they are...
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