Combining Matrix Transformations (AQA GCSE Further Maths): Revision Note

• If a question asks you to prove a geometric fact about combined transformations "using matrix multiplication", you cannot just draw a sequence of diagrams for your answer

  • you must write each transformation as a matrix and use QP or PQ (depending on the order)

Three transformations in the $x$-$y$ plane are given below.

$\mathbf{A}=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$  represents an enlargement by scale factor -1 about the origin
$\mathbf{B}=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$ represents a reflection in the y-axis
$\mathbf{C}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ represents a reflection in the x-axis

Use matrix multiplication to prove that A is the same as B followed by C.
 

Transformation $\mathbf{B}$ followed by transformation $\mathbf{C}$ would be combined into a single matrix by finding $\mathbf{CB}$ (note the order)

Find the matrix multiplication $\mathbf{CB}$

$\mathbf{CB}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\times \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}\left(1\times -1+0\times 0\right)& \left(1\times 0+0\times 1\right)\\ \left(0\times -1+-1\times 0\right)& \left(0\times 0+-1\times 1\right)\end{array}\right)$

Simplifying, it can be seen that this is the same as $\mathbf{A}$

$\mathbf{CB}\mathbf{=}\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\mathbf{=}\mathbf{A}$

This makes sense geometrically as well: a reflection in the y-axis then the x-axis is equivalent to an enlargement of scale factor -1 (the same as a rotation of 180° about the origin)