MODULE 12

MATEMATIK SPM ENRICHMENT

TOPIC : MATRICES

TIME : 2 HOURS

3

2

4

n

1.

(a)

The inverse matrix of

is m

5

4

5

3

Find the value of m and of n.

(b) Hence, using matrices, solve the following simultaneous equations : 3x 2y = 8

5x 4y = 13

Answer :

(a)

(b)

2.

(a)

Given that G =

m

3

and the inverse matrix of G is

1

4

3

,

2

n

14

2

m

find the value of m and of n.

Hence, using matrices, calculate the value of p and of q that satisfies the following equation :

p

1

G

q

8

Answer :

(a)

(b)

3.

(a)

Given that

1

2

1

0

find matrix A.

A

,

3

5

0

1

Hence, using the matrix method, find the value of r and s which satisfy the simultaneous equations below.

-r + 2s = -4 -3r + 5s = -9

Answer :

(a)

(b)

4

5

1

0

4.

Given matrix P =

and matrix PQ =

6

8

0

1

Find the matrix Q. Hence, calculate by using the matrix method, the values of m and n that

satisfy the following simultaneous linear equations : 4m + 5n = 7

6m + 8n = 10

Answer :

(a)

(b)

4

3

5.

Given the matrix P is

,

8

5

1

0

(a)

Find the matrix Q so that PQ =

0

1

(b)

Hence, calculate the values of h and k, which satisfy the matrix equation:

4 3h 7

8

5

k

11

Answer :

(a)

(b)

k

6

6.

(a)

Given matrix M =

, find the value of k if matrix M has no inverse.

4

2

(b)

Given the matrix equations

7 6 x 4

and

x 1 8 6 4

5

8

y

1

y

h

5

7

1

Find the value of h

Hence, find the value of x and y.

Answer :

(a)

(b)

2

5

7.

It is given that matrix P =

does not have an inverse matrix.

k

2

(a)

Find the value of k.

(b)

If k = 1, find the inverse matrix of P and hence, using matrices, find the

values of x and y that satisfy the following simultaneous linear equations.

2x + 5y = 13 x - 2y = -7

Answer :

(a)

(b)

8.

(a)

Find matrix M such that

2

4

2

4

M

1

3

1

3

Using matrices, calculate the values of x and y that satisfy the following matrix equation.

2

4

x

6

1

3

y

5

Answer :

(a)

(b)

9.

(a)

Find the inverse of matrix

3 1

.

5

2

Hence, using matrices, calculate the values of d and e that satisfy the following simultaneous equations :

2d e = 7

5d e = 16

Answer :

(a)

(b)

1

2

10.

Given matrix M =

, find

2

5

(a)

the inverse matrix of M

(b)

hence, using matrices, the values of u and v that satisfy the following

simultaneous equations :

u 2v = 8 2u + 5v = 7

Answer :

(a)

(b)

MODULE 12 - ANSWERS

TOPIC : MATRICES

1.

(a)

m =

1

1m

2

n = 2

1m

(b)

3

2 x

8

=

1m

5

4

y

13

x

1

4

28

1m

y

5

2

3

13

x = 3

1m

y =

1

1m

2

2.

(a)

n = 4

1m

m = 5

1m

(b)

5

3 p

1

2

4

q

8

p

1

4

3

1

q

14

2

5

8

p = 2

q = -3

5

2

3.

(a)

A =

3

1

1 2 r 4

(b)

3

5

s

9

1m

1m

1m

1m

2m

1m

r 1 5 2 4

1m

s

1

3

1

9

r = -2

1m

s = -3

8

5

1m

4.

(a)

P =

1

1m

32 30

6

4

1

8

5

=

2

6

4

1m

4

5m

7

(b)

1m

6

8

n

10

m

1

8

5 7

1m

n

2

6

4

10

m = 3

1m

n = -1

1m

5.

(a)

P

1

5

3

1m

20 (24)

8

4

1

5

3

1m

4

8

4

(b)

4 3 h 7

8

5

k

11

h

1

5 3 7

1m

k

2

8

4

11

1

2

1m

2

100

h = 1

1m

k = -50

1m

6.

(a)

k = -12

1m

(b)

(i)

h = 26

1m

x

1

8

6 4

y

26

5

7

1

1

26

26

13

(ii)

x = -1

y =

1

2

7.

(a)

- 4 5k = 0

1m

5k = -4

k =

4

1m

5

(b)

2

5 x

13

1

2

y

7

x

1

2

5 13

y

9

1

2

7

x = -1

y = 3

8.

(a)

M =

1

0

0

1

(b)

x

1

3

46

y

6 4

1

2

5

1

3

4 6

2

1

2

5

1

2

2

4

x = -1

y = 2

1m

1m

1m

1m

1m

1m

1m

1m

2m

1m

1m

1m

1m

9.

(a)

1

2

1

6 5

5

3

1

2

1

1

5

3

(b)

2

1d

7

5

3

e

16

d

1

3

1 7

e

1

5

2

16

1

5

3

1

5

3

d = 5

e = 3

10.

(a)

1

5

2

5 (4)

2

1

1

5

2

9

2

1

(b)

1

2u

8

2

5

v

7

u

1

5

28

v

9

2

1

7

1 54

9

9

6

1

u 6

v 1

1m

1m

1m

1m

1m

1m

1m

1m

1m

1m

1m

1m