## Transformation Matrices - Double Checking my Answers

A transformation T is given by:

$\displaystyle T: \begin{pmatrix} x \\ y\end{pmatrix}\rightarrow\begin{pmatrix} x+2y \\ y-x\end{pmatrix}$

(i). Write down the matrix representing the transformation $\displaystyle T$.
(ii). The transformation S is an anticlockwise rotation through $\displaystyle 90^{\circ}$ about the origin.
Find the matrix representing the transformation $\displaystyle S$.
(iii). Find the single matrix representing the transformation $\displaystyle S$ followed by the transformation $\displaystyle T$.

(i). $\displaystyle T: \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x+2y \\ y-x \end{pmatrix}\Longrightarrow$ $\displaystyle \left\{ \begin{gathered} ax + by = x+2y \hfill \\ cx+dy=y-x \hfill \\ \end{gathered} \right.$
$\displaystyle \therefore\begin{array}{rcrcrc}a=1\\b=2\\c=-1\\d=1\end{array}$
(ii). $\displaystyle S = \begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}$
(iii). $\displaystyle TS = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}=\begin{pmatrix} 3 & 3 \\ 0 & 3 \end{pmatrix}$