MATHS

1. 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
 1
4

−
2 m

   
find the value of m and of n.
(2) Hence, using matrices, calculate the value of p and of q that
satisfies the following equation :
 p  
1


2. G
 =


q
  −
8
  
Answer :
(a)
(b)
3
. (a)
Given
that

−1
2
 1 0 
find matrix
A.
A

 =


,

−
3 5
 
0 1

   
(2) 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)

3. (b)

4
5


1
0

4
. Given matrix P =   and matrix PQ = 

6 8
 
0 1  
(1) Find the matrix Q.
(2) 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. 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 − 3h  − 7




 =


8
−
5


k


−
1
1
   
Answer :
(a)
(b)
k
6

6
. (a) Given matrix M =  , find the value of k if matrix M has no inverse.

2


5. −
4
 
(b)
Given the matrix
equations
 7 − 6 x  −
4  a
n
d
 x  1 8 6 − 4



 =
  

=  
 
 −
5 8


y
 
1
 
y

h

5 7


1



      

 
(1) Find the value of h
(2) 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 =

6. 13 x -
2y = -7
Answer :
(a)
(b)
8
. (a)
Find matrix M such
that

2 4 

2
4

 M =


1 3
 
1 3
  
(2) 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)

7. (b)

8. 9
. (a)
Find the inverse of
matrix
3 −1

.



5
−
2

 
(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)

9. 
1
− 2

1
0. 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


10. 



=

 1
m
5
−
4


y
 
1
3



   

x

1
−
4 28 
1m
 
=

 

y
 
−
5
 
2
3
1
3
    
x =
3 1m
y
=
−
1
1m
2
2
.
(
a
)
n
= 4
1
m
m
= 5
1
m
(
b
)

5 3  p  
1





=  

2 4


q

 −
8




  
 p  1 4
−
3

1



=



q

1
4
−
2 5

 −
8
 


p =
2
q = -3

5
−
2
3
.
(
a
) A =  

3
−
1

 
−1 2 r   − 4



 =

(b
)

−
3 5


s
  −
9


  
1
m
1
m
1
m
1
m
2
m
1
m
r  1 5 − 2 −



=
 s  1 3 r =
-2
1
m

11. s =
-3
81 
4
. (a)
P
=

12. 1

8
−
5

=
2

−
6 4
1
m


4
5m


7

(b
)
 

 =
 1
m

6 8


n
 1
0


 

m

1
8
− 5 7

 
=



1
m

n

2
−
6 4


1
0

   
m =
3
1
m
n =
-1
1
m
5
.
(a
) P
1
−
5
3

1
m
=  
− 20 −
(−24)

8 4

 
1
 −
5
3

1
m= 

4

8 4



(b
)
4 − 3 h   −
7 



 =


8
−
5


k


−
1
1
   

h

1
− 5 3 − 7

   



13. =
1
m

k

2

8 4

 −
1
1

   

1
 2
=  1
m
2

−1
00

h =
1
1
m
k =
-50
1
m
6
.
(a
)
k =
-12
1
m
(b
)
(
i
)
h =
26
1
m
 x 
1

8
6 −
4 


=
 



y
 2
6

5 7


1

  

 
1
−
26
=  
2
6

−
1
3

 
(
i
i
)
x =
-1
y
=
−
1
2

14. 7
.
(a
)
- 4 – 5k
= 0
1
m
5k = -4
k
= −
4 1
m
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
−
46


=
 


y
 6
−
4

−
1 2


5
  


1

3
− 4
6
=



2
−
1 2


5

 
1
− 2

= 
4 

15. 2

x =
-1
y =
2
1m
1m
1m
1m
1m
1m
1m
1m
2m
1m
1m
1m
1m
9.
(
a
) 1

−
2
1

 

16. − 6
+ 5

−
5 3

 
1
−
2
1

=  
−
1

−
5 3

 
(
b
)

2
−1d

 7

 

 =
 

5
−
3


e
 
1
6



   

d

1
−
3
1 
7 
 
=


e

−
1
−
5 2


1
6
 

1
−
5
= 

−
3−1


5

=
 

3

 
d =
5
e =
3
10
.
(
a
)
1
 5
2

 
5 −
(−4)

−
2
1

 

17. 1
 5
2

=  
9

−
2 1

 
(
b
)

1
−
2u 

8

 

 =



2 5


v
 
7



   

u

1
 5
28

 
=


 

v 9

−
2 1


7

 
 
1 
54 
= 
9

−
9

 6

=
 

−
1

 
u =
6
v =
−1
1m
1m
1m

18. 1m
1m
1m
1m
1m
1m
1m
1m
1m