8.
$$\mathbf { P } = \left( \begin{array} { r r }
3 a & - 4 a
4 a & 3 a
\end{array} \right) , \text { where } a \text { is a constant and } a > 0$$
- 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 ( \(- 20 a , 15 a\) ). - Find the coordinates of the vertices of \(T _ { 1 }\)
- 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 \(0 < \alpha < \frac { \pi } { 2 }\)
- 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 \(W\).
- Find the matrix \(\mathbf { R }\).