IGCSE Further Maths – Matrices
Exercise 1 – Matrix Multiplication (questions from AQA Worksheets)
Question 1
Work out
(a)
 4 2  7 

  
  3 5  1
(b)
5 0   3

 

0 5   4
(c)
 5  2

2 
6  3
(d)
 1 0  3 

  
 0 1   2
(e)
 4 7 

6 
  1  3
(f)
8 4   3

  
 4 2  6 
Question 2
 2  1

A = 
3 4 
 2 3 

C = 
 1  1
7 4

B = 
5 3
Work out
(a)
AB
(b)
BC
(c)
3A
(d)
BA
(e)
C
(f)
 1  4

B 
 5 7 
Question 3
  2 0

P = 
 5 1
 4 1 

Q = 
 3  2
 3 
C =  
  2
Work out
(a)
P2
(b)
QP
(c)
5Q
(d)
PC
(e)
IQ
(f)
3I
Question 4
Work out
(a)
 2  1  0 3 

 

 1 3   1  4
(b)
  3  2   2 4

 

  1 5   3 4
(c)
3 2  5  2

 

7 5   7 3 
(d)
 10  7   2 4 



8    2 3 
9
(e)
 1  2  2 3 


 
3  5  1 4
(f)
 2 3   1  2

 

 1 4 3  5
Question 5 (Non-calculator)
Work out, giving your answers as simply as possible.
(a)
 2
1   2
0 

 1 3 2   3  2 2 



(b)
 1

 2
 3

 2

 1


5

  2 4




 1 3


 2

(c)
 3 2


7 5
2
3 3

 2

(d)
 4   3 1 
3 3    4 0 
1

3
2

3
(e)
1

2
1

4
(f)
 2 3


1 4
 2

 7

2 
3 
2
Question 6
Work out, giving your answers as simply as possible.
 p 


 p  1
(a)
1 0 


 0  1
(d)
2 0   a

 
0 2  0
0

a 
(b)
3 0  x 

  
0 3  y 
(c)
 0 1  m 

  
 1 0   2m 
(e)
 4t 0   3 0 


 
 0 4t   0 3 
(f)
1 0 


 0  1
1 0   3 

  
 0 1   2 
Question 7
Work out, giving your answers as simply as possible.
(a)
 2 x  3   x 3x 

 

  5 4x    3 0 
(b)
8
 a 3a   7

 

  2 1    10 11
(c)
 x 0


1 x
(d)
 y

 3
(e)
a  a 1
a 
a 1

 

 a  2 a  1   a  2 a  1
(f)
 3x  3 


  9 x  1
y

x 
2 3y


0 1 
Exercise 1b
Question 1
Question 3
Question 2
2
−1 0
0 ) (𝑎
𝑏
a) Work out ( 1
)
𝑐
3
Give your answer in terms of 𝒂, 𝒃, 𝒄.
2 −1 0 𝑏
b) If ( 1 0 ) (
) = 𝑰 where 𝑰 is the identity
𝑎 𝑐
3
matrix, work out the values of 𝑎, 𝑏, 𝑐.
Question 4
2
2
Exercise 2
1. [Jan 2013 Paper 2 Q15] Describe fully the
single transformation represented by the
0 −1
matrix (
)
1 0
2. [Set 2 Paper 1 Q4] The transformation matrix
𝑎 2
(
) maps the point (3,4) onto the
−1 1
point (2, 𝑏). Work out the values of 𝑎 and 𝑏.
𝑎
𝑏
3. [Set 3 Paper 1 Q6] The matrix (
)
−𝑎 2𝑏
maps the point (5,4) onto the point (1,17).
Work out the values of 𝑎 and 𝑏.
4. [Worksheet 2 Q5] Work out the image of the
point D (1, 2) after transformation by the
2 3
matrix (
)
−1 1
5. [Worksheet 2 Q6] The point A(m, n) is
transformed to the point A (2, 0) by the
2 3
matrix (
)
1 1
Work out the values of m and n.
6. [Worksheet 2 Q8] Describe fully the
transformation given by the matrix
0 −1
(
)
−1 0
7. [Worksheet 2 Q9] The unit square OABC is
ℎ 0
transformed by the matrix (
) to the
0 ℎ
square OABC.
The area of OABC is 27. Work out the exact
value of h.
8. [Specimen Paper 2 Q20] (a) Matrix 𝑨 =
4 3
(
)
1 1
Work out the image of point 𝑃(2, −1) using
transformation matrix 𝑨.
(b) Point 𝑄 is (0,1)
Line 𝑃𝑄 is transformed to line 𝑃′𝑄′ using
matrix 𝑨.
Work out the length of 𝑃′𝑄′.
Exercise 3
1. Point (3, −2) is transformed by the matrix
1 −1
(
) followed by a further
0 1
0 2
transformation by the matrix (
).
1 0
(i) Work out the matrix for the combined
transformation.
(ii) Work out the co-ordinates of the image
point of 𝑃.
2. Point (−1,4) is transformed by the matrix
3 −1
(
) followed by a further
−2 2
1 0
transformation by the matrix (
).
3 −2
(i) Work out the matrix for the combined
transformation.
(ii) Work out the co-ordinates of the image
point of 𝑊.
3. The unit square is reflected in the 𝑥-axis
followed by a rotation through 180° centre
the origin. Work out the matrix for the
combined transformation.
4. The unit square is enlarged, centre the origin,
scale factor 2 followed by a reflection in the
line 𝑦 = 𝑥. Work out the matrix for the
combined transformation.
−1 0
5. [Jan 2013 Paper 2 Q17] (
) represents
0 1
0 1
a reflection in the 𝑦-axis. (
) represents
1 0
a reflection in the line 𝑦 = 𝑥.
Work out the matrix that represents a
reflection in the 𝑦-axis followed by a
reflection in the line 𝑦 = 𝑥.
6. [June 2012 Paper Q22] The transformation
0 −1
matrix (
) maps a point 𝑃 to 𝑄. The
−1 0
1 0
transformation matrix (
) maps point
0 −1
𝑄 to point 𝑅.
Point 𝑅 is (−4,3). Work out the coordinates
of point 𝑃.
7. [Set 1 Paper Q14b] The unit square OABC is
transformed by reflection in the line 𝑦 = 𝑥
followed by enlargement about the origin
with scale factor 2. What is the matrix of the
combined transformation?
3 0
−1 0
8. 𝐴 = (
) and 𝐵 = (
).
0 3
0 1
The point 𝑃(2,7) is transformed by matrix 𝐵𝐴
to 𝑃′. Show that 𝑃′ lies on the line
7𝑥 + 2𝑦 = 0.