AQA Level 2 Further Maths · Level 2 Certificate · Chapter 9: Matrices · Lesson 1

Multiplying matrices and the identity matrix

About 75 minutesNo calculator74 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve the simultaneous equations $4 x + 3 y = 7$ and $2 x - y = 11$ .
[2 marks]
Answer $x = \dots\dots y = \dots\dots$
A2
Needed today
Solve $x^{2} - 2 x = 15$ .
[2 marks]
Answer
A3
From chapter 8
Work out the gradient of the curve $y = 2 x^{3} - 5 x$ at the point where $x = - 1$ .
[2 marks]
Answer
A4
Last lesson
Work out the coordinates of the stationary point on the curve $y = 3 x^{2} - 12 x + 5$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

Example, then your turn

Level 2

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

Example 1

$A = (31 - 2 4)$ and $v = (52)$ . Work out $2 A$ and $Av$ .

Hint To multiply by a number, multiply every entry. For

Av

, multiply each row of

A

by the column.

B1
$A = (- 1 4 32)$ and $v = (2 - 3)$ . Work out $3 A$ and $Av$ .
[3 marks]

Example 2

$P = (2 - 3 14)$ and $Q = (15 - 2 3)$ . Work out $PQ$ and $QP$ .

Hint Row

i

of the first matrix times column

j

of the second gives the entry in row

i

, column

j

B2
$M = (32 - 1 5)$ and $N = (4 - 1 02)$ . Work out $MN$ and $NM$ .
[4 marks]

Example 3

$A = (23 - 1 1)$ and $v = (4 - 1)$ . (a) Work out $A^{2}$ . (b) Explain why $vA$ cannot be worked out.

Hint

A^{2}

means

AA

. A product needs the number of columns of the first to equal the number of rows of the second.

B3
$B = (1 - 2 43)$ and $w = (3 - 1)$ . (a) Work out $B^{2}$ . (b) One of $Bw$ and $wB$ can be worked out. Work it out.
[4 marks]

Example 4

Given that $(21 a 3) (b - 1 42) = (6 - 1 410)$ , work out the values of $a$ and $b$ .

Hint Multiply out the left-hand side in terms of

a

and

b

. Then equate the entries that hold only one unknown first.

B4
Given that $(3 - 1 p 2) (q 2 1 - 4) = (122 - 9 - 9)$ , work out the values of $p$ and $q$ .
[4 marks]
Answer $p = \dots\dots q = \dots\dots$

Example 5

The identity matrix is $I = (1001)$ . Given that $(4131) (a c b d) = I$ , work out the values of $a$ , $b$ , $c$ and $d$ .

Hint For any

2 \times 2

matrix

A

AI = IA = A

. Equate the first column to get two equations in

a

and

c

, then the second column for

b

and

d

B5
Given that $(5332) (p r q s) = I$ , where $I$ is the identity matrix, work out the values of $p$ , $q$ , $r$ and $s$ .
[5 marks]
Answer $p = \dots q = \dots r = \dots s = \dots$

Practice

Level 2

C1
$A = (12 k 3)$ and $B = (3447)$ . Work out the value of $k$ for which $AB = BA$ .
[4 marks]
Answer $k = \dots\dots$
C2
$M = (x 1 5 x)$ . Given that $M^{2} = (14 - 6 - 30 14)$ , work out the value of $x$ .
[4 marks]
Answer $x = \dots\dots$

C3
Given that $(35 - 2 4) (x y) = (86)$ , work out the values of $x$ and $y$ .
[4 marks]
Answer $x = \dots\dots y = \dots\dots$
C4
$A = (32 4 - 3)$ . Show that $A^{2} = k I$ , where $k$ is a number, and state the value of $k$ . Hence work out $A^{4}$ .
[4 marks]

Exam-style questions

AQA exam style

D1
Leo works out $(31 2 - 2) (4 - 1 15)$ . He writes $(12 - 1 2 - 10)$ .
Explain Leo's mistake. Work out the correct answer.
[3 marks]

D2
$(20 c 5) (a 0 - 3 2) = k I$ , where $k$ is a constant and $I$ is the identity matrix. Work out the values of $a$ , $c$ and $k$ .
[4 marks]
Answer $a = \dots\dots c = \dots\dots k = \dots\dots$
D3
$M = (k 2 31)$ , where $k$ is a positive number.
(a)
Work out $M^{2}$ . Give your answer in terms of $k$ .
[2 marks]
(b)
The top left entry of $M^{2}$ is $22$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
(c)
Use your value of $k$ to show that $M^{2} - 5 M = 2 I$ .
[2 marks]
(d)
Hence work out the matrix $N$ such that $MN = I$ .
[2 marks]
Total for D3: 8 marks
D4
Given that $(x 2 1 x) (x 3) = (19 y)$ , work out the two possible pairs of values of $x$ and $y$ .
[4 marks]
Answer $x = \dots\dots, y = \dots\dots$ or $x = \dots\dots, y = \dots\dots$
D5
$M$ is a $2 \times 2$ matrix such that $(2513) M = (41112)$ . Work out $M$ .
[5 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$M$ is a $2 \times 2$ matrix. Every entry of $M$ is a positive whole number. $M^{2} = (7934)$ . Work out $M$ .
[6 marks]
Hint 1 · What to try
Let $M = (a c b d)$ and work out $M^{2}$ in terms of $a$ , $b$ , $c$ and $d$ .
Hint 2 · The first line
The two diagonal entries give $a^{2} + b c = 7$ and $d^{2} + b c = 4$ . Subtract them: $a^{2} - d^{2} = 3$ , so $(a - d) (a + d) = 3$ .
Hint 3 · The full method
With whole numbers, $a + d = 3$ and $a - d = 1$ , so $a = 2$ and $d = 1$ . Then $3 b = 3$ and $3 c = 9$ , so $M = (2311)$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply a 2 by 2 matrix by a number, by a column and by another 2 by 2 matrix.
I know that AB and BA are usually different, and when a product cannot be worked out.
I can find unknown entries by equating elements, including when the answer is kI.
I can find a matrix M with AM = I.

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