Cambridge Additional Mathematics

312 Matrices (Chapter 12)

8aFor A=

μ

¡ 10

15

¶

, B=

μ

34

¡ 1 ¡ 2

¶

, and C=

μ

4 ¡ 1

¡ 13

¶

, find (A+B)+C

and A+(B+C).

b Prove that, ifA,B, andCare any 2 £ 2 matrices, then (A+B)+C=A+(B+C).

Hint: Let A=

μ

ab

cd

¶

, B=

μ

pq

rs

¶

, and C=

μ

wx

yz

¶

.

If A=(aij) has order m£n, andkis a scalar, then kA=(kaij).

So, to findkA, we multiply each element inAbyk.

The result is another matrix of order m£n.

Example 4 Self Tutor

IfAis

μ

125

201

¶

, find:

a 3 A b^12 A

a 3 A=3

μ

125

201

¶

=

μ

3 £ 13 £ 23 £ 5

3 £ 23 £ 03 £ 1

¶

=

μ

3615

60 3

¶

b^12 A=^12

μ

125

201

¶

=

Ã

1

2 £^1

1

2 £^2

1

2 £^5

1

2 £^2

1

2 £^0

1

2 £^1

!

=

à 1

2 12

1

2

(^1012)
!
We use capital letters for
matrices and lower-case
letters for scalars.

MULTIPLES OF MATRICES

In the pantry there are 6 cans of peaches, 4 cans of apricots, and 8 cans of pears. We represent this by the

column vector C=

0

@

6

4

8

1

A.

If we doubled the cans in the pantry, we would have

0

@

12

8

16

1

A which is C+C or 2 C.

Notice that to get 2 CfromCwe simply multiply all the matrix elements by 2.

Likewise, trebling the fruit cans in the pantry gives 3 C=

0

@

3 £ 6

3 £ 4

3 £ 8

1

A=

0

@

18

12

24

1

A

and halving them gives^12 C=

0

B

@

1

2 £^6

1

2 £^4

1

2 £^8

1

C

A=

0

@

3

2

4

1

A.

cyan magenta yellow black

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100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_12\312CamAdd_12.cdr Tuesday, 7 January 2014 5:56:21 PM BRIAN