1. The document contains 10 mathematics word problems involving matrix operations and inverses. 2. The problems require finding inverse matrices, solving systems of equations using matrices, and calculating values that satisfy matrix equations. 3. Detailed step-by-step solutions are provided for each problem.

1. PPR Maths nbk
MODULE 12
SKIM TUISYEN FELDA (STF) MATEMATIK SPM “ENRICHMENT”
TOPIC : MATRICES
TIME : 2 HOURS
⎛3 − 2⎞ ⎛− 4 n⎞
1. (a) The inverse matrix of ⎜
⎜ ⎟ is m
⎟ ⎜
⎜− 5 ⎟
⎝5 − 4⎠ ⎝ 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. PPR Maths nbk
Given that G = ⎜
⎛ m 3⎞ 1 ⎛ 4 − 3⎞
2. (a)
⎜ ⎟ and the inverse matrix of G is ⎜ ⎟,
⎝ 2 n⎟
⎠ 14 ⎜ − 2 m ⎟
⎝ ⎠
find the value of m and of n.
(b) 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. PPR Maths nbk
⎛ −1 2⎞ ⎛1 0⎞
3. (a) Given that A⎜
⎜ − 3 5 ⎟ = ⎜ 0 1 ⎟,
⎟ ⎜ ⎟
find matrix A.
⎝ ⎠ ⎝ ⎠
(b) 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. PPR Maths nbk
⎛ 4 5⎞ ⎛1 0⎞
Given matrix P = ⎜ ⎟ and matrix PQ = ⎜
⎜0 1⎟
4.
⎜ 6 8⎟ ⎟
⎝ ⎠ ⎝ ⎠
(a) Find the matrix Q.
(b) 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)

5. PPR Maths nbk
⎛ 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 ⎟ = ⎜ − 11⎟
⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
Answer :
(a)
(b)

6. PPR Maths nbk
⎛ 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 ⎞ ⎛ x⎞ 1 ⎛ 8 6 ⎞⎛ − 4 ⎞
⎜
⎜ − 5 8 ⎟⎜ y ⎟ = ⎜ 1 ⎟
⎟⎜ ⎟ ⎜ ⎟ and ⎜ ⎟ = ⎜
⎜ y ⎟ h ⎜ 5 7 ⎟⎜ 1 ⎟
⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
(i) Find the value of h
(ii) Hence, find the value of x and y.
Answer :
(a)
(b)

7. PPR Maths nbk
⎛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. PPR Maths nbk
⎛ 2 4⎞ ⎛ 2 4⎞
Find matrix M such that ⎜ ⎟M = ⎜
⎜1 3⎟
8. (a)
⎜ 1 3⎟ ⎟
⎝ ⎠ ⎝ ⎠
(b) 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. PPR Maths nbk
⎛3 −1⎞
9. (a) Find the inverse of matrix ⎜ ⎟ .
⎜ 5 − 2⎟
⎝ ⎠
(b) 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)

10. PPR Maths nbk
⎛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)

11. PPR Maths nbk
MODULE 12 - ANSWERS
TOPIC : MATRICES
1
1. (a) m= − 1m
2
n =2 1m
(b) ⎛ 3 − 2⎞ ⎛ x ⎞ ⎛8 ⎞
⎜ ⎟⎜ ⎟ = ⎜ ⎟
⎜5 − 4 ⎟ ⎜ y ⎟ ⎜13 ⎟
1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ x ⎞ 1 ⎛− 4 2 ⎞⎛ 8 ⎞ 1m
⎜ ⎟= ⎜
⎜ y⎟ 2 ⎜− 5 ⎟⎜ ⎟
⎝ ⎠ ⎝ 3 ⎟⎜13 ⎟
⎠⎝ ⎠
x=3 1m
1
y= − 1m
2
2. (a) n =4 1m
m=5 1m
(b) ⎛ 5 3 ⎞⎛ p ⎞ ⎛ 1 ⎞
⎜
⎜2 ⎟⎜ ⎟ = ⎜ ⎟
⎜ − 8⎟
4 ⎟⎜ q ⎟
1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ p ⎞ 1 ⎛ 4 − 3 ⎞⎛ 1 ⎞
⎜ ⎟= ⎜
⎜ q ⎟ 14 ⎜ − 2 5 ⎟⎜ − 8 ⎟
⎟⎜ ⎟
1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
p=2 1m
q = -3 1m
⎛5 − 2⎞
3. (a) A=⎜ ⎟ 2m
⎜3 −1 ⎟
⎝ ⎠
⎛ −1 2⎞ ⎛ r ⎞ ⎛ − 4⎞
(b) ⎜− 3 5⎟ ⎜ s ⎟ = ⎜ − 9⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟ 1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ r ⎞ 1 ⎛ 5 − 2 ⎞⎛ − 4 ⎞
⎜ ⎟= ⎜
⎜ s ⎟ 1 ⎜ 3 − 1 ⎟⎜ − 9 ⎟
⎟⎜ ⎟ 1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
r = -2 1m
s = -3 1m
1 ⎛ 8 − 5⎞
4. (a) P= ⎜ ⎟ 1m
32 − 30 ⎜ − 6 4 ⎟
⎝ ⎠

12. PPR Maths nbk
1 ⎛ 8 − 5⎞
⎜ ⎟
=
2 ⎜− 6 4 ⎟
⎝ ⎠
1m
⎛ 4 5 ⎞⎛ m ⎞ ⎛ 7 ⎞
(b) ⎜
⎜ 6 8 ⎟⎜ n ⎟ = ⎜10 ⎟
⎟⎜ ⎟ ⎜ ⎟ 1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ m ⎞ 1 ⎛ 8 − 5 ⎞⎛ 7 ⎞
⎜ ⎟= ⎜
⎜ n ⎟ 2 ⎜ − 6 4 ⎟⎜10 ⎟
⎟⎜ ⎟ 1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
m=3 1m
n = -1 1m
1 ⎛ − 5 3 ⎞ 1m
5. (a) P = ⎜ ⎟
− 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 ⎞
⎜ ⎟= ⎜
⎜ k ⎟ 2 ⎜ 8 4 ⎟⎜ − 11⎟
⎟⎜ ⎟ 1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
1⎛ 2 ⎞
= ⎜ ⎟
2 ⎜ − 100 ⎟
1m
⎝ ⎠
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 ⎟
⎝ ⎠

13. PPR Maths nbk
(ii) 1m
1m
x = -1 1m
1
y= − 1m
2
7. (a) - 4 – 5k = 0 1m
5k = -4
4
k= − 1m
5
(b) ⎛ 2 5 ⎞⎛ x ⎞ ⎛ 13 ⎞
⎜
⎜ 1 − 2 ⎟⎜ y ⎟ = ⎜ − 7 ⎟
⎟⎜ ⎟ ⎜ ⎟ 1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ x⎞ 1 ⎛ − 2 − 5 ⎞⎛ 13 ⎞
⎜ ⎟=− ⎜
⎜ y⎟ ⎟⎜ ⎟
9 ⎜ − 1 2 ⎟⎜ − 7 ⎟
1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
x = -1 1m
y=3 1m
⎛1 0⎞
8. (a) M= ⎜ ⎟ 2m
⎜ ⎟
⎝0 1⎠
(b) ⎛ x⎞ 1 ⎛ 3 − 4 ⎞⎛ 6 ⎞
⎜ ⎟=
⎜ y ⎟ 6 − 4 ⎜ − 1 2 ⎟⎜ 5 ⎟
⎜ ⎟⎜ ⎟ 1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
1 ⎛ 3 − 4 ⎞⎛ 6 ⎞
= ⎜ ⎟⎜ ⎟
2 ⎜ − 1 2 ⎟⎜ 5 ⎟
⎝ ⎠⎝ ⎠
1 ⎛ − 2⎞
= ⎜ ⎟ 1m
2⎜ 4 ⎟
⎝ ⎠
x = -1 1m
y=2 1m

14. PPR Maths nbk
9. (a) 1 ⎛ − 2 1⎞
⎜ ⎟
− 6 + 5 ⎜ − 5 3⎟
1m
⎝ ⎠
1 ⎛ − 2 1⎞
= ⎜ ⎟
− 1 ⎜ − 5 3⎟
1m
⎝ ⎠
(b) ⎛ 2 − 1 ⎞⎛ d ⎞ ⎛ 7 ⎞
⎜
⎜ 5 − 3 ⎟⎜ e ⎟ = ⎜16 ⎟
⎟⎜ ⎟ ⎜ ⎟ 1m
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ d ⎞ 1 ⎛ − 3 1 ⎞⎛ 7 ⎞
⎜ ⎟=
⎜ e ⎟ − 1 ⎜ − 5 2 ⎟⎜16 ⎟
⎜ ⎟⎜ ⎟ 1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
1 ⎛ − 5⎞
= ⎜ ⎟
− 1 ⎜ − 3⎟
⎝ ⎠
⎛ 5⎞
=⎜ ⎟
⎜ 3⎟
⎝ ⎠
d=5 1m
e=3 1m
10. (a) 1 ⎛ 5 2⎞
⎜ ⎟
5 − (−4) ⎜ − 2 1 ⎟
1m
⎝ ⎠
1 ⎛ 5 2⎞
= ⎜ ⎟
9 ⎜− 2 1⎟
1m
⎝ ⎠
(b) ⎛ 1 − 2 ⎞⎛ u ⎞ ⎛ 8 ⎞
⎜ ⎜ 2 5 ⎟⎜ v ⎟ = ⎜ 7 ⎟
1m
⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ u ⎞ 1 ⎛ 5 2 ⎞⎛ 8 ⎞
⎜ ⎟= ⎜
⎜ v ⎟ 9 ⎜ − 2 1 ⎟⎜ 7 ⎟
⎟⎜ ⎟
1m
⎝ ⎠ ⎝ ⎠⎝ ⎠
1 ⎛ 54 ⎞
= ⎜ ⎟
9 ⎜ − 9⎟
⎝ ⎠
⎛6⎞
=⎜ ⎟
⎜ − 1⎟
⎝ ⎠
u=6 1m
v = −1 1m