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c.

[ ]

=

 y x

1

2

5d.

 y x

 y x

313

4

8

3

2

0

6

0

1

2

3

0

2

In problem 3 -5, compute the required matrices if:

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314211

316

,423101

251

,513241

,2014

,1121

03EDCBA

3. a) D + E b) 2B - C c) 4E - 2D d) -3(D + 2E)

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4. a) 2AT + C b) (D-E)T c) BT d)  AC T 

4

1

2

1−

5. a) AB b) BA c) (3E)D d) (AB)C e) DAT

6. Evaluate i) determinant ii) minor matrix iii) cofactor matrix iv) adjoin matrix

v) inverse of the matrix, for each the following matrix:

a) A =

25

13b) B =

− 12

46c)

2 5 5

1 1 0

2 4 3

C 

= − −

d)

2 0 3

0 3 2

2 0 4

 D

= − −

7. Solve the following system by using:

i. Inverse method

ii. Cramer’s rule

a) 2x + y = 4 b) 3p + 2q = 1 .

x - 2y = 7 2q - p = -3

c) x1 + x2 + x3 = 6 d) s + 2t + 3m = 14

x1 - x2 + x3 = 2 3s + t + 2m = 114x1 + 2x3 = 14 2s + 3t + m = 11

8 An automobile manufacturer produces different types of vehicles in different models. In

the matrix below, the rows represent, from top to bottom, sedans, vans and trucks, and the

columns represent, from left to right, standard, custom and stock models. Each entry is a

suggested dealer price.

500,20700,22800,19

500,30000,31600,29

400,22500,23000,21

 RM   RM   RM  

 RM   RM   RM  

 RM   RM   RM  

What is the suggested dealer price for:(a) a stock model van (b) a stock model truck (c) a custom model sedan

 

9 Three components A,B and C are used in the manufacturing of a product. For a

 production run, 100, 150, and 200 units of A, B, and C are RM2, RM1, and RM3,

respectively. Represent the total cost of A, B, and C by a matrix product.

10. Let matrix A represent the sales (in thousands of dollars) of a toy company in 1998 in

three cities, and let B represent the sales in the same cities in 2000, where A and B are given

 by

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A =

850280450

150350400

Action ,

Educational

B =

750320460

220330380

Action

Educational

10.

If the company buys a competitor and doubles its 2000 sales in 2001, what is the change

in sales between 1998 and 2001?

11. The following table shows the wholesale price in RM per kg, of three commodities,

namely red chillies, long beans, and cucumber, in the three towns of Kuala Terengganu,

Kuala Lumpur and Johor Bahru.

Commodity

Town

Red

Chillies

Long

 beans

Cucumber 

Kuala Terengganu 4 2 2

Kuala Lumpur 2 2 2

Johor Bahru 4 4 2

A company with branches located in each of the three towns has been awarded a contract in

each town to supply  x kg red chillies,  y kg long beans, and  z kg cucumber in each town to the

local retailer. The vegetables are obtained from the town itself. The profit earned by the

 branches in Kuala Terengganu, Kuala Lumpur, and Johor Bahru respectively are RM3000,RM2000, and RM3500.

(a) Write a matrix equation in  x,  y, and  z to represent the above information.

(b) Determine the quantities of red chillies, long beans and cucumber supplied in each town

using cramer’s rule.

ANSWERS TUTORIAL 1

1 (a)

3 4 5

5 6 77 8 9

 A

=

(b)

3 3

1 6

2 1

3 2

 B

=

 

2 (a) a = 3, b = 1 (b) x = 2, y = 0, z = 1 (c) x = 3, y = 6 (d) x = 5,y = 2

3 (a)

7 6 5

2 1 3

7 3 7

−

(b) Not defined (c)

22 6 8

2 4 6

10 0 4

− −

(d)

39 21 24

9 6 15

33 12 30

− − − − − − − −

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4 (a)7 2 4

3 5 7

(b)

5 0 1

4 1 1

1 1 1

− − − − −

(c)4 0

1 2

−

(d)

1 61

9 04

3 9

−

5 (a)

12 3

4 5

4 1

− −

(b) Not defined (c)

42 108 75

12 3 21

36 78 63

−

(d)

3 45 9

11 11 17

7 17 13

−

(e) Not

defined

6 (a) (i) 1 (ii)2 5

1 3

(iii)2 5

1 3

− −

(iv)2 1

5 3

− −

(v)2 1

5 3

− −

(b) (i) 14 (ii)

1 2

4 6

−

(iii)

1 2

4 6

− (iv)

1 4

2 6

−

(v)

1 41

2 614

−

(c) (i) –1 (ii)

3 3 2

5 4 2

5 5 3

− − − − − −

(iii)

3 3 2

5 4 2

5 5 3

− − − −

(iv)

3 5 5

3 4 5

2 2 3

− − − −

(v)

3 5 5

3 4 5

2 2 3

− − − − −

(d) (i) –6 (ii)

12 4 6

0 2 0

9 4 6

− − −

(iii)

12 4 6

0 2 0

9 4 6

− − − − −

(iv)

12 0 9

4 2 4

6 0 6

− − − − −

 

(v)

12 0 91

4 2 46

6 0 6

− − − − − −

7 (a) x = 3, y = -2 (b) p = 1, q = -1 (c) 1 2 33, 2, =1  x x x= = (d) s = 1, t = 2, m = 3

8 (a) RM30,500 (b) RM20,500 (c) RM23,500

9. RM950

10

=−

650360470

2903103602 B A

11. (a)

244

222

224

 Z 

Y  

 X  

=

3500

2000

3000

(b) x=500 kg, y = 250 kg, z = 250 kg

6