## Transformation Matrices - Double Checking my Answers

A transformation T is given by:

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

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

(i). $T: \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x+2y \\ y-x \end{pmatrix}\Longrightarrow$ $\left\{ \begin{gathered}
ax + by = x+2y \hfill \\
cx+dy=y-x \hfill \\
\end{gathered} \right.$

$\therefore\begin{array}{rcrcrc}a=1\\b=2\\c=-1\\d=1\end{array}$

(ii). $S = \begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}$

(iii). $TS = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}=\begin{pmatrix} 3 & 3 \\ 0 & 3 \end{pmatrix}$

Are these answers correct? I'm fairly sure about part 1, but not too sure about part 2 as I wasn't 100% sure on how to work it out. If part 2 is wrong, part 3 will also be wrong, but I'd only like assistance with part 2 .