Edexcel IGCSE Math B 5.G Combined transformations Question Bank

Practise combining transformation matrices, choosing the correct order, and using inverses to find a single equivalent matrix.

Exam points

form the combined matrix in the correct order when one transformation follows another
use an inverse matrix to find an unknown transformation between two mapped shapes
apply a product matrix to coordinates before drawing the final image on a grid

[Maximum number: 3]

$\mathbf A=\begin{pmatrix}2k^2&k-9\\-3k&k+1\end{pmatrix}, \qquad \mathbf B=\begin{pmatrix}1&-5\\3&k\end{pmatrix}.$
The determinant of matrix $\mathbf A$ is equal to the determinant of matrix $\mathbf B$ .

The transformation with matrix $\mathbf C$ , where $\mathbf C$ is a $2\times2$ matrix, is equivalent to the transformation with matrix $\mathbf A$ followed by the transformation with matrix $\mathbf B$ .
Given that k is positive, find matrix $\mathbf C$ .

$\text{Determinant of matrix }\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc.$

Positive root:
k=3.
Substituting:
$A= \begin{pmatrix} 18 & -6\\ -9 & 4 \end{pmatrix}, \qquad B= \begin{pmatrix} 1 & -5\\ 3 & 3 \end{pmatrix}.$
Since A is followed by B,
C=BA.
Final answer:
$C= \begin{pmatrix} 63 & -26\\ 27 & -6 \end{pmatrix}.$

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5.G Combined transformations question 1