Problem

$\mathbf{P} = \begin{pmatrix} \frac{1}{2} & -\frac{3}{2} \\ \frac{3}{2} & \frac{1}{2} \end{pmatrix}$

The matrix $\mathbf{P}$ represents the transformation $U$

(a) Give a full description of $U$ as a single geometrical transformation.

The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf{Q}$, is a reflection in the line $y = -x$

(b) Write down the matrix $\mathbf{Q}$

The transformation $U$ followed by the transformation $V$ is represented by the matrix $\mathbf{R}$

(c) Determine the matrix $\mathbf{R}$

The transformation $W$ is represented by the matrix $3\mathbf{R}$

The transformation $W$ maps a triangle $T$ to a triangle $T'$

The transformation $W'$ maps the triangle $T'$ back to the original triangle $T$

(d) Determine the matrix that represents $W'$