M Matrix Transformations (all calculations restricted to 2x2 or 2x1 matrices) — AQA 8365 spec coverage

M1

Multiplication of matrices (2x2 by 2x2 or by 2x1), and multiplication by a scalar

A scalar multiplies every entry of a matrix.
For matrix multiplication, combine each row of the first matrix with each column of the second using multiply-then-add.
The product $AB$ is defined when the number of columns of $A$ equals the number of rows of $B$ ; here calculations use only $2\times2$ and $2\times1$ matrices.
Matrix multiplication is order-sensitive: in general $AB\ne BA$ .

Tier 1 · Easy

1 mark

ORIGINAL

Work out $3\begin{pmatrix}2&-1\\0&4\end{pmatrix}$ .

Tier 2 · Standard

2 marks

ORIGINAL

Work out $\begin{pmatrix}2&-1\\3&4\end{pmatrix}\begin{pmatrix}5\\-2\end{pmatrix}$ .

Tier 3 · Hard

4 marks

ORIGINAL

Given $\begin{pmatrix}p&2\\1&q\end{pmatrix}\begin{pmatrix}3&-1\\2&4\end{pmatrix}=\begin{pmatrix}13&5\\9&11\end{pmatrix}$ , work out $p$ and $q$ .

M2

The identity matrix I (2x2 only)

The $2\times2$ identity matrix is $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ .
It leaves a compatible matrix unchanged: $AI=IA=A$ .
For a column vector, multiplying by $I$ leaves the represented point unchanged.
Do not confuse the identity matrix with the zero matrix; its diagonal entries are $1$ .

Tier 1 · Easy

1 mark

ORIGINAL

Write down the $2\times2$ identity matrix $I$ .

Tier 2 · Standard

2 marks

ORIGINAL

Let $A=\begin{pmatrix}2&-3\\5&1\end{pmatrix}$ . Work out both $AI$ and $IA$ .

Tier 3 · Hard

3 marks

ORIGINAL

Let $P=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ . Show that $P^2=I$ and hence work out $P^{17}$ .

M3

Transformations of the unit square in the x-y plane, represented by a 2x2 matrix (rotations of 90/180/270 about the origin, reflections in x=0, y=0, y=x, y=-x, enlargements centred on the origin)

The columns of a transformation matrix are the images of $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ respectively.
Standard matrices should be linked to rotations about the origin, reflections in $x=0$ , $y=0$ , $y=x$ or $y=-x$ , and enlargements centred at the origin.
To transform a point $(x,y)$ , multiply the matrix by the column vector $\begin{pmatrix}x\\y\end{pmatrix}$ .
When describing a transformation, give its type and all defining details: angle and direction, mirror line, or scale factor and centre.

Tier 1 · Easy

2 marks

ORIGINAL

Write down the matrix for a rotation of $90^\circ$ anticlockwise about the origin.

Tier 2 · Standard

3 marks

ORIGINAL

Describe fully the transformation represented by $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ and work out the image of $(4,-3)$ .

Tier 3 · Hard

4 marks

ORIGINAL

A transformation maps $\begin{pmatrix}1\\0\end{pmatrix}$ to $\begin{pmatrix}0\\-1\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ to $\begin{pmatrix}-1\\0\end{pmatrix}$ . Write its matrix, describe the transformation fully, and find the image of $(3,-2)$ .

M4

Combination of transformations using matrix multiplication

If matrix $A$ acts first and matrix $B$ acts second, the combined matrix is $BA$ .
Multiply transformation matrices using the usual row-by-column rule; order matters because generally $AB\ne BA$ .
A combined matrix may simplify to a familiar single transformation, which should be described fully when requested.
A common error is to write matrices in the order transformations are stated rather than in reverse action order.

Tier 1 · Easy

2 marks

ORIGINAL

A reflection in the $x$ -axis is followed by a reflection in the $y$ -axis. Work out the combined matrix.

Tier 2 · Standard

3 marks

ORIGINAL

Matrix $A$ rotates points $90^\circ$ anticlockwise about the origin. Matrix $B$ reflects points in the $x$ -axis. Transformation $A$ acts before transformation $B$ . Work out the combined matrix and describe its effect.

Tier 3 · Hard

4 marks

ORIGINAL

A point is reflected in $y=x$ , enlarged by scale factor $3$ about the origin, then rotated $90^\circ$ clockwise about the origin. Work out the combined matrix, describe its geometric effect, and find the image of $(-2,1)$ .