Combining Matrix Transformations (Edexcel IGCSE Maths)

Combining Transformation Matrices

Spec point — 5G

How do I find a single matrix that represents a combination of transformations?

• A point (x,y) can be transformed twice

  • First by the matrix $P$, then second by the matrix $Q$
  • This is called a combined (or composite) transformation

• A single matrix, $M$, representing the combined transformation can be found using matrix multiplication as follows:

  • $M=\mathrm{QP}$

    • The order matters: the first transformation is the last in the multiplication

    • The order is the reverse of what you may expect!

      • The first transformation is on the right, with the second transformation to its left

  • $\mathrm{PQ}$ would represent $Q$ first, followed by $P$

Exam Hint 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)

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

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

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

Answer:

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

Find the matrix multiplication $\mathrm{CB}$

$\begin{gathered} \text{CB} & = ( \begin{gathered} 1 0 \\ 0 - 1 \end{gathered} ) \times ( \begin{gathered} - 1 & 0 \\ 0 1 \end{gathered} ) \\ = ( \begin{gathered} 1 \times ( - 1 ) + 0 \times 0 & 1 \times 0 + 0 \times 1 \\ 0 \times ( - 1 ) + ( - 1 ) \times 0 & 0 \times 0 + ( - 1 ) \times 1 \end{gathered} ) \\ = ( \begin{gathered} - 1 + 0 & 0 + 0 \\ 0 + 0 & 0 - 1 \end{gathered} ) \\ = ( \begin{gathered} - 1 & 0 \\ 0 - 1 \end{gathered} ) \end{gathered}$

This is the same as $A$

• 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)

$\text{CB} = ( \begin{gathered} - 1 & 0 \\ 0 - 1 \end{gathered} ) = \mathrm{A}$