Cambridge Additional Mathematics

Answers 489

EXERCISE 12B.3

1a 3 A b O c ¡C dO e 2 A+2B

f¡A¡B g ¡ 2 A+C h 4 A¡B i 3 B

2aX=A¡B bX=C¡B c X=2C¡ 4 B

dX=^12 A eX=^13 B f X=A¡B

gX=2C hX=^12 B¡A i X=^14 (A¡C)

3aX=

³

36

918

́

bX=

μ 1

2 ¡

1

4

3

4

5

4

¶

cX=

μ

¡ 1 ¡ 6

1 ¡^12

¶

EXERCISE 12C.1

1a(11) b(22) c (16) 2b(wxyz)

0

B

B

B

@

1 4 1 4 1 4 1 4

1

C

C

C

A

3aP=

¡

27 35 39

¢

, Q=

Ã

4

3

2

!

btotal cost=

¡

27 35 39

¢

Ã

4

3

2

!

=$ 291

4aP=

¡

10 6 3 1

¢

, N=

0

@

3

2

4

2

1

A

btotal points=

¡

10631

¢

0

@

3

2

4

2

1

A=56points

EXERCISE 12C.2

1 Number of columns inAdoes not equal number of rows inB.

2am=n b 2 £ 3 cBhas 3 columns, Ahas 2 rows

3adoes not exist b(28 29)

4a(8) b

Ã

20 3

8012

40 6

!

5a(3 5 3) b

Ã

¡ 2

1

1

!

6aQ=

Ã

32 24

25 16

13 9

!

bP=

³ 1 : 19

1 : 55

́

cQP=

Ã

32 24

25 16

13 9

!

³ 1 : 19

1 : 55

́

=

Ã

75 : 28

54 : 55

29 : 42

!

It represents the total value of sales for each pen colour.

d$ 75 :28 +$ 54 :55 +$ 29 :42 =$ 159 : 25

7aC=

³

12 : 5

9 : 5

́

N=

³

2375 5156

2502 3612

́

b

³

78 669: 5

65 589

́

income from day 1

income from day 2 c $144 258:^50

8aR=

Ã

11

12

23

!

bP=

³ 7319

6222

́

c

³

48 70

52 76

́

di$ 48 ii $ 76

eThe elements ofPRtell us that, if all the items are to be

bought at one store, it is cheapest to do so at store A for

both you and your friend. However, the cheapest way is to

buy paint from store A, and hammers and screwdrivers from

store B.

EXERCISE 12C.3

1aA^2 +A b B^2 +2B cA^3 ¡ 2 A^2 +A

dA^3 +A^2 ¡ 2 A eAC+AD+BC+BD

fA^2 +AB+BA+B^2 gA^2 ¡AB+BA¡B^2

hA^2 +2A+I i 9 I¡ 6 B+B^2

2aA^3 =3A¡ 2 I, A^4 =4A¡ 3 I

bB^3 =3B¡ 2 I, B^4 =6I¡ 5 B, B^5 =11B¡ 10 I

cC^3 =13C¡ 12 I, C^5 = 121C¡ 120 I

3a iI+2A ii 2 I¡ 2 A iii 10 A+6I

bA^2 +A+2I ci¡ 3 A ii ¡ 2 A iii A

4aA^2 =

μ 1

2

1

2

1

2

1

2

¶

bfalse as A(A¡I)=O does not imply that

A=O orA¡I=O

c

³ 00

00

́

,

³ 10

01

́

,

Ã

ab

a¡a^2

b

1 ¡a

!

, b 6 =0

5 For example, A=

³

01

00

́

gives A^2 =

³

00

00

́

.

6aa=3, b=¡ 4 ba=1, b=8

7ap=¡ 2 ,q=1 bA^3 =5A¡ 2 I

cA^4 =¡ 12 A+5I

EXERCISE 12D.1

1a

³ 30

03

́

=3I,

μ

1 ¡ 2

¡^2353

¶

b

³

10 0

010

́

=10I,

³

0 : 20 : 4

¡ 0 : 10 : 3

́

2a¡ 2 b¡ 1 c 0 d 1

3a 26 b 6 c ¡ 1 da^2 +a

4a¡ 3 b¡ 3 c ¡ 12 5 Hint: LetA=

³

ab

cd

́

6a idetA=ad¡bc iidetB=wz¡xy

iii AB=

³

aw+by ax+bz

cw+dy cx+dz

́

iv detAB=(ad¡bc)(wz¡xy)

7adetA=¡ 2 , detB=¡ 1

bidet(2A)=¡ 8 iidet (¡A)=¡ 2

iii det(¡ 3 B)=¡ 9 iv det (AB)=2

8a 141

³

5 ¡ 4

12

́

b

³

10

1 ¡ 1

́

c does not exist

d

³

10

01

́

e 101

³

20

15

́

f does not exist

g¡ 151

³

7 ¡ 2

¡ 4 ¡ 1

́

h 101

³

2 ¡ 4

13

́

i

³

¡ 3 ¡ 1

21

́

9a

1

2 k+6

³

2 ¡ 1

6 k

́

, k 6 =¡ 3 b

1

3 k

³

k 1

03

́

, k 6 =0

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