Problem

(i)

$\mathbf{P} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

The matrix $\mathbf{P}$ represents a geometrical transformation $U$

(a) Describe $U$ fully as a single geometrical transformation.

The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf{Q}$, is a rotation through $240°$ anticlockwise about the origin followed by an enlargement about $(0, 0)$ with scale factor $6$

(b) Determine the matrix $\mathbf{Q}$, giving each entry in exact numerical form.

Given that $U$ followed by $V$ is the transformation $T$, which is represented by the matrix $\mathbf{R}$

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

(ii) The transformation $W$ is represented by the matrix

$\begin{pmatrix} -2 & 3 \\ 2 & -2 \end{pmatrix}$

Show that there is a real number $\lambda$ for which $W$ maps the point $(\lambda, 1)$ onto the point $(4\lambda, 4)$, giving the exact value of $\lambda$