Cambridge Additional Mathematics

Matrices (Chapter 12) 315

2 FindXin terms ofA,B, andCif:

a X+B=A b B+X=C c 4 B+X=2C

d 2 X=A e 3 X=B f A¡X=B

g^12 X=C h 2(X+A)=B i A¡ 4 X=C

3aSuppose M=

μ

12

36

¶

and^13 X=M. FindX.

b Suppose N=

μ

2 ¡ 1

35

¶

and 4 X=N. FindX.

c Suppose A=

μ

10

¡ 12

¶

, B=

μ

14

¡ 11

¶

, and A¡ 2 X=3B. FindX.

Suppose you go to a shop and purchase 3 cans of soft drink, 4 chocolate bars, and 2 ice creams.

The prices are:

soft drink cans

$ 1 : 30

chocolate bars

$ 0 : 90

ice creams

$ 1 : 20

We can represent this by the quantities matrix A=

¡

342

¢

and the costs matrix B=

0

@

1 : 30

0 : 90

1 : 20

1

A.

We can find the total cost of the items by multiplying the number of each item by its respective cost, and

then adding the results:

3 £$ 1 :30 + 4£$ 0 :90 + 2£$ 1 :20 =$ 9 : 90

We can also determine the total cost by thematrix multiplication:

AB=

¡

342

¢

0

@

1 : 30

0 : 90

1 : 20

1

A

=(3£ 1 :30) + (4£ 0 :90) + (2£ 1 :20)

=9: 90

Notice that we write therow matrixfirst and thecolumn matrixsecond.

In general,

¡

abc

¢

0

@

p

q

r

1

A=ap+bq+cr.

EXERCISE 12C.1

1 Determine:

a

¡

3 ¡ 1

¢

μ

5

4

¶

b

¡

132

¢

0

@

5

1

7

1

A c

¡

6 ¡ 123

¢

0

B

@

1

0

¡ 1

4

1

C

A

C Matrix multiplication

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_12\315CamAdd_12.cdr Tuesday, 7 January 2014 5:56:43 PM BRIAN