AQA Level 2 Further Maths · Level 2 Certificate · Chapter 9: Matrices

Multiplying matrices, transformations and combining them

About 80 minutesNo calculator106 marks

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Date

Do Now

The first is what this chapter needs. The rest bring back earlier work so nothing is forgotten.

A1
Needed today
Write down the image of the point $(3, - 6)$ after (a) a reflection in the line $y = x$ (b) a rotation of $18 0^{\circ}$ about $O$ .
[2 marks]
Answer(a) $(\dots, \dots)$ (b) $(\dots, \dots)$

A2
From chapter 8
$y = 2 x^{3} - 5 x^{2} + 4$ . Work out the gradient of the curve at the point where $x = 2$ .
[2 marks]
Answer
A3
From chapter 6
Solve $2 cos x = - 1$ for $0^{\circ} ⩽ x ⩽ 36 0^{\circ}$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A4
From chapter 4
Solve the simultaneous equations $3 x + 2 y = 4$ and $x - y = 3$ .
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
A5
GCSE Higher
Expand and simplify $(2 x - 3) (x + 4) - x (x + 5)$ .
[2 marks]
Answer

Examples, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

$M = (a 2 3 b)$ and $M^{2} = (10697)$ .

Work out the values of $a$ and $b$ .

Hint Multiply

M

by itself and match each entry. Use the two equations with squares, then the one that decides the signs.

B1
$N = (p 3 - 2 q)$ and $N^{2} = (- 5 6 - 4 3)$ .
Work out the values of $p$ and $q$ .
[4 marks]
Answer $p = \dots\dots$ $q = \dots\dots$

Example 2

Triangle $T$ is reflected in the $x$ -axis. The image is then reflected in the line $y = x$ .

(a) Work out the matrix that represents the combined transformation.

(b) Describe fully the single transformation it represents.

Hint The first transformation is on the right: (reflection in

y = x

)

\times

(reflection in the

x

-axis).

B2
Triangle $T$ is reflected in the $y$ -axis. The image is then rotated $9 0^{\circ}$ anticlockwise about $O$ .
(a) Work out the matrix that represents the combined transformation.
(b) Describe fully the single transformation it represents.
(c) Work out the image of the vertex $(4, 1)$ .
[4 marks]
Answer(a) $(\dots \dots \dots \dots)$ (c) $(\dots, \dots)$

Example 3

The matrix $M$ maps the point $(2, 1)$ to $(4, - 1)$ and the point $(1, 3)$ to $(7, 2)$ .

Work out $M$ .

Hint Let

M = (a c b d)

. Each point gives one equation for the top row and one for the bottom row.

B3
The matrix $M$ maps the point $(1, 2)$ to $(4, 5)$ and the point $(3, 1)$ to $(7, 0)$ .
Work out $M$ .
[4 marks]
Answer $(\dots \dots \dots \dots)$

Example 4

$A = (k 1 2 - 1)$ . The point $(2, 1)$ is transformed by $A$ , and its image is transformed by $A$ again. The final image is $(26, 7)$ .

Work out the value of $k$ .

Hint Work out the first image, then the second, in terms of

k

. One coordinate gives a quadratic with two roots; the other coordinate decides.

B4
$A = (k 2 - 1 1)$ . The point $(1, 3)$ is transformed by $A$ , and its image is transformed by $A$ again. The final image is $(5, 9)$ .
Work out the value of $k$ .
[4 marks]
Answer $k = \dots\dots$

Practice

Level 2

Each question asks for something different. Show your working.

C1
$A = (2 - 1 34)$ and $B = (05 - 2 1)$ . Work out $AB$ and $BA$ .
[3 marks]
Answer $AB =$ $(\dots \dots \dots \dots)$ $BA =$ $(\dots \dots \dots \dots)$

C2
$A = (3 - 4 2 - 3)$ . Show that $A^{2} = I$ . Hence write down $A^{5}$ .
[3 marks]
Answer $A^{5} =$ $(\dots \dots \dots \dots)$
C3
$(a 3 2 b) (a 1) = (117)$ . Work out the two possible pairs of values of $a$ and $b$ .
[3 marks]
Answer $a = \dots$ , $b = \dots$ or $a = \dots$ , $b = \dots$
C4
Write down the matrix that represents (a) a reflection in the line $y = - x$ (b) an enlargement, scale factor $5$ , centre $O$ .
[2 marks]
Answer(a) $(\dots \dots \dots \dots)$ (b) $(\dots \dots \dots \dots)$
C5
Describe fully the single transformation represented by the matrix $(0 - 1 10)$ .
[2 marks]
C6
The unit square $O A B C$ is transformed by the matrix $(3012)$ .
Draw the image of the unit square on the grid, and work out its area.
[3 marks]
Answerarea $= \dots\dots$
C7
The matrix $(k - 2 3 k)$ maps the point $(2, - 1)$ to the point $(5, - 8)$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
C8
$R = (01 - 1 0)$ represents a rotation of $9 0^{\circ}$ anticlockwise about $O$ .
Work out $R^{3}$ . Describe fully the transformation it represents.
[2 marks]
Answer $R^{3} =$ $(\dots \dots \dots \dots)$
C9
The matrix $(3211)$ maps the point $P$ to the point $(7, 5)$ . Work out the coordinates of $P$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

Exam-style questions

AQA exam style

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$A = (2111)$ and $B = (1 - 1 - 1 2)$ .
(a)
Work out $AB$ .
[2 marks]
Answer $(\dots \dots \dots \dots)$
(b)
The point $Q$ is the image of a point under $A$ . Hence explain what $B$ does to $Q$ .
[1 mark]
(c)
The point $P$ is mapped to $(8, 5)$ by $A$ . Use $B$ to work out the coordinates of $P$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D1: 5 marks

D2
Triangle $T$ is reflected in the line $y = - x$ to give $T_{1}$ . $T_{1}$ is then enlarged, scale factor $3$ , centre $O$ , to give $T_{2}$ .
(a)
Write down the matrix $P$ for the reflection and the matrix $Q$ for the enlargement.
[2 marks]
Answer $P =$ $(\dots \dots \dots \dots)$ $Q =$ $(\dots \dots \dots \dots)$
(b)
Work out the single matrix that maps $T$ to $T_{2}$ .
[2 marks]
Answer $(\dots \dots \dots \dots)$
(c)
Work out the image in $T_{2}$ of the vertex $(5, 1)$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(d)
Work out the area of $T_{2}$ .
[1 mark]
Answer $\dots\dots$ square units
Total for D2: 6 marks
D3
Ben says, "A rotation of $18 0^{\circ}$ about $O$ followed by an enlargement, scale factor $- 2$ , centre $O$ , is an enlargement with scale factor $- 2$ ."
Ben is wrong. Use matrices to show that he is wrong, and describe fully the single transformation.
[3 marks]
D4
$M = (3212)$ .
(a)
Given that $M^{2} = p M + q I$ , work out the values of $p$ and $q$ .
[3 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
(b)
Hence write $M^{3}$ in the form $a M + b I$ .
[2 marks]
Answer $M^{3} = \dots\dots M + \dots\dots I$
Total for D4: 5 marks
D5
$M = (0 k k 0)$ , where $k > 0$ . The unit square $O A B C$ is transformed by $M$ . The area of its image is $49$ .
(a)
Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
(b)
$M$ represents a combination of two transformations. Describe them fully.
[2 marks]
(c)
Work out the image of the point $(- 2, 3)$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
Total for D5: 5 marks
D6
The matrix $(p q q p)$ maps the point $(2, 5)$ to the point $(11, 17)$ .
(a)
Work out the values of $p$ and $q$ .
[2 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
(b)
Hence work out the coordinates of the point that is mapped to $(8, 8)$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D6: 4 marks
D7
A shape is reflected in the $y$ -axis, and the image is then reflected in the line $y = - x$ .
(a)
Work out the matrix for the combined transformation, and describe fully the single transformation it represents.
[3 marks]
(b)
The two reflections are now done in the other order. Describe fully the single transformation.
[2 marks]
Total for D7: 5 marks
D8
Triangle $A B C$ has vertices $A (3, 3)$ , $B (9, 3)$ and $C (3, 6)$ . It is transformed by the matrix $(- \frac{1}{3} 0 0 - \frac{1}{3})$ .
(a)
Work out the coordinates of the image of $B$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(b)
Describe fully the transformation the matrix represents.
[1 mark]
(c)
Work out the area of the image of triangle $A B C$ .
[2 marks]
Answer $\dots\dots$ square units
Total for D8: 4 marks
D9
$A = (1031)$ .
(a)
Work out $A^{2}$ and $A^{3}$ .
[2 marks]
Answer $A^{2} =$ $(\dots \dots \dots \dots)$ $A^{3} =$ $(\dots \dots \dots \dots)$
(b)
Hence write down $A^{n}$ in terms of $n$ .
[1 mark]
Answer $(\dots \dots \dots \dots)$
(c)
$A^{n}$ maps the point $(1, 1)$ to the point $(61, 1)$ . Work out the value of $n$ .
[2 marks]
Answer $n = \dots\dots$
Total for D9: 5 marks

Extension

Stretch

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
$M = (1 k k 1)$ , where $k > 1$ . The unit square $O A B C$ is transformed by $M$ . The area of its image is $8$ .
Work out the value of $k$ .
[4 marks]
Hint 1 · What to try
Work out the images of $A (1, 0)$ , $B (1, 1)$ and $C (0, 1)$ . The image is a parallelogram.
Hint 2 · The first line
The images are $(1, k)$ , $(1 + k, 1 + k)$ and $(k, 1)$ . Draw the square from $(0, 0)$ to $(1 + k, 1 + k)$ round it and take away the two regions outside the parallelogram.
Hint 3 · The full method
Each region outside splits into two triangles of area $\frac{1}{2} (1 + k)$ , so the parallelogram has area $(1 + k)^{2} - 2 (1 + k) = k^{2} - 1$ . Then $k^{2} - 1 = 8$ , so $k = 3$ .
Answer $k = \dots\dots$

E2
$A = (a c b - a)$ . $A^{2} = I$ , and $A$ maps the point $(1, 1)$ to the point $(5, - 3)$ .
Work out $A$ .
[4 marks]
Hint 1 · What to try
Multiply $A$ by itself. The top-left entry gives $a^{2} + b c = 1$ .
Hint 2 · The first line
The image of $(1, 1)$ gives $a + b = 5$ and $c - a = - 3$ , so $b = 5 - a$ and $c = a - 3$ .
Hint 3 · The full method
Substitute: $a^{2} + (5 - a) (a - 3) = 1$ , which is $8 a - 15 = 1$ , so $a = 2$ , $b = 3$ , $c = - 1$ .
Answer $(\dots \dots \dots \dots)$
E3
$T$ is a transformation. A reflection in the line $y = x$ followed by $T$ is the same as an enlargement, scale factor $- 5$ , centre $O$ .
Work out the matrix for $T$ . Describe $T$ fully as a combination of two transformations.
[4 marks]
Hint 1 · What to try
Let $Q$ be the matrix for the reflection. Then $TQ = - 5 I$ .
Hint 2 · The first line
$Q^{2} = I$ , because reflecting twice puts every point back. Multiply both sides on the right by $Q$ : $T = - 5 Q$ .
Hint 3 · The full method
$T = (0 - 5 - 5 0) = (5005) (0 - 1 - 1 0)$ .
Answer $T =$ $(\dots \dots \dots \dots)$
E4
$M = (01 - 1 1)$ .
Show that $M^{6} = I$ . Hence work out $M^{100}$ .
[4 marks]
Hint 1 · What to try
Work out $M^{2}$ , then $M^{3} = M M^{2}$ .
Hint 2 · The first line
$M^{3} = (- 1 0 0 - 1) = - I$ , so $M^{6} = (- I)^{2} = I$ .
Hint 3 · The full method
$100 = 6 \times 16 + 4$ , so $M^{100} = M^{4} = M^{3} M = - M$ .
Answer $M^{100} =$ $(\dots \dots \dots \dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply matrices, use the identity matrix, and find unknown entries from a product or a power.
I can transform points and the unit square with a matrix, find the matrix for a transformation, and describe what a matrix does.
I can combine transformations by multiplying their matrices in the right order, and describe the single transformation.

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