Matrix Transformations (AQA GCSE Further Maths): Exam Questions

Here are two transformations.

A    Rotation $90°$ clockwise about the origin.
B    Reflection in the line $y=x$

Use matrix multiplication to work out the single matrix which represents the combined transformation A followed by
B.

The transformation matrix $\left(\begin{array}{cc}a& b\\ 2a& 3b\end{array}\right)$ maps the point $\left(1,-3\right)$ onto the point $\left(1,4\right)$

Work out the values of $a$ and $b$.

You must show your working.

$a=$..........................   $b=$..........................

Use matrix multiplication to show that, in the $x–y$ plane,

• a reflection in the line $y=-x$, followed by

• a rotation, $90°$ anticlockwise about the origin, followed by

• a reflection in the $x$-axis

is equivalent to a transformation by the identity matrix.

The transformation matrix $\left(\begin{array}{cc}2a& b\\ -b& -a\end{array}\right)$ maps the point $\left(3,4\right)$ onto the point $\left(8,-7\right)$

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

$a=...................,b=...................$

Use matrix multiplication to show that, in the $x–y$ plane,

• a reflection in the line $x=0$, followed by

• a rotation of $90°$ anticlockwise, centre the origin,

is equivalent to a single transformation, which you should describe.

Under the transformation represented by $\left(\begin{array}{cc}-1& -3\\ 2& 4\end{array}\right)$,

the image of point $P\left(a,2\right)$ is point $Q$.

Can point $Q$ be the same as point $P$?

You must show your working.

Shape $A$ maps to shape $B$ by an enlargement, scale factor $3$, centre the origin.

Shape $B$ maps to shape $C$ by a rotation through $180°$, centre the origin.

Shape $A$ can be mapped to shape $C$ by a single transformation.

Use matrices to show that the single transformation is an enlargement, centre the origin.

State the scale factor of the enlargement.

Use matrices to prove that

a reflection in the line $y=-x$ followed by a rotation of $90°$ anticlockwise about the origin

is not the same as

a rotation of $90°$ anticlockwise about the origin followed by a reflection in the line $y=-x$

In each case, state the single transformation that they represent.