8.

$$\mathbf { P } = \left( \begin{array} { r r } 
3 a & - 4 a \\
4 a & 3 a
\end{array} \right) , \text { where } a \text { is a constant and } a > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the matrix $\mathbf { P } ^ { - 1 }$ in terms of $a$.\\
(3)

The matrix $\mathbf { P }$ represents the transformation $U$ which transforms a triangle $T _ { 1 }$ onto the triangle $T _ { 2 }$.\\
The triangle $T _ { 2 }$ has vertices at the points ( $- 3 a , - 4 a$ ), ( $6 a , 8 a$ ), and ( $- 20 a , 15 a$ ).
\item Find the coordinates of the vertices of $T _ { 1 }$
\item Hence, or otherwise, find the area of triangle $T _ { 2 }$ in terms of $a$.

The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a rotation through an angle $\alpha$ clockwise about the origin, where $\tan \alpha = \frac { 4 } { 3 }$ and $0 < \alpha < \frac { \pi } { 2 }$
\item Write down the matrix $\mathbf { Q }$, giving each element as an exact value.

The transformation $U$ followed by the transformation $V$ is the transformation $W$. The matrix $\mathbf { R }$ represents the transformation $W$.
\item Find the matrix $\mathbf { R }$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel F1 2015 Q8 [13]}}