Combining Matrix Transformations (Edexcel IGCSE Maths B): 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 should write each transformation as a matrix

• and combine them using 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.

Answer:

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 $\mathbf{CB}$

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

This is the same as $\mathbf{A}$

• This makes sense geometrically as well

  • A reflection in the y-axis followed by a reflection in the x-axis

  • is equivalent to an enlargement of scale factor -1 (which is the same as a rotation of 180° about the origin)

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