Cambridge Additional Mathematics

Matrices (Chapter 12) 331

11 Solve using an inverse matrix:

a

½

3 x¡ 4 y=2

5 x+2y=¡ 1

b

½

4 x¡y=5

2 x+3y=9

12 Suppose A=2A¡^1.

a Show that A^2 =2I.

b Simplify (A¡I)(A+3I), giving your answer in the form rA+sI whererandsare

real numbers.

Review set 12B

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1 For P=

0

@

12

10

23

1

A and Q=

0

@

30

14

11

1

A, find:

a P+Q b Q¡P c^32 P¡Q

2 A library owns several copies of a popular trilogy of

novels, according to the matrix:

a At present, the books on loan are described by the

matrix B=

0

@

20

11

32

1

A. Write a matrix to describe

the books currently on the shelves.

b The values of the books (in dollars) are described by the matrix C=

μ

778

15 16 20

¶

.

i Which book has value $ 16?

ii Find the total value of the books currently on loan.

3 Prove that for any square matrixA, AO=OA=O.

4 WriteXin terms ofAandBif:

a 2 X=B¡A b 3(A+X)=2B c B¡ 4 X=A

5 Suppose A=

μ

31

¡ 10

¶

, B=

μ

¡ 22

1 ¡ 3

¶

, and A+2X=¡B. FindX.

6 IfAis

¡

123

¢

andBis

0

@

24

01

32

1

A, find, if possible:

a 2 B b^12 B c AB d BA

7 IfAandBare square matrices, under what conditions are the following true?

a If AB=B then A=I. b (A+B)^2 =A^2 +2AB+B^2

paperback hard cover

A=

0

@

42

52

63

1

A

book 1

book 2

book 3

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Y:\HAESE\CAM4037\CamAdd_12\331CamAdd_12.cdr Wednesday, 8 January 2014 9:44:51 AM BRIAN