Cambridge Additional Mathematics

Matrices (Chapter 12) 329

b In matrix form, the system is

μ

23

54

¶μ

x

y

¶

=

μ

4

17

¶

which has the form AX=B.

c Premultiplying by A¡^1 , A¡^1 AX=A¡^1 B

) X=A¡^1 B

=¡^17

μ

¡ 35

14

¶

=

μ

5

¡ 2

¶

) x=5and y=¡ 2.

EXERCISE 12E

1 Convert into matrix equations:

a

½

3 x¡y=8

2 x+3y=6

b

½

4 x¡ 3 y=11

3 x+2y=¡ 5

c

½

3 a¡b=6

2 a+7b=¡ 4

2 Use matrix algebra to solve the system:

a

½

2 x¡y=6

x+3y=14

b

½

5 x¡ 4 y=5

2 x+3y=¡ 13

c

½

x¡ 2 y=7

5 x+3y=¡ 2

d

½

3 x+5y=4

2 x¡y=11

e

½

4 x¡ 7 y=8

3 x¡ 5 y=0

f

½

7 x+11y=18

11 x¡ 7 y=¡ 11

3aShow that if AX=B then X=A¡^1 B, whereas if XA=B then X=BA¡^1.

b FindXif:

i

μ

¡ 65

¡ 34

¶

X=

μ

3 ¡ 2

01

¶

ii X

μ

12

5 ¡ 1

¶

=

μ

14 ¡ 5

22 0

¶

iii

μ

13

2 ¡ 1

¶

X=

μ

1 ¡ 3

42

¶

iv X

μ

24

3 ¡ 1

¶

=

μ

810

¡ 515

¶

4aConsider the system

½

2 x¡ 3 y=8

4 x¡y=11

.

i Write the equations in the form AX=B, and find detA.

ii Does the system have a unique solution? If so, find it.

b Consider the system

½

2 x+ky=8

4 x¡y=11

.

i Write the system in the form AX=B, and find detA.

ii For what value(s) ofkdoes the system have a unique solution? Find the unique solution.

iii Findkwhen the system does not have a unique solution. How many solutions does the system

have in this case?

)

μ

x

y

¶

=¡^17

μ

4 ¡ 3

¡ 52

¶μ

4

17

¶

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_12\329CamAdd_12.cdr Friday, 4 April 2014 5:11:07 PM BRIAN