Find inverse transformation matrix

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a \\ 4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

1. 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 ( \(- 20 a , 15 a\) ).
2. Find the coordinates of the vertices of \(T _ { 1 }\)
3. 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 \(0 < \alpha < \frac { \pi } { 2 }\)
4. 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 \(W\).
5. Find the matrix \(\mathbf { R }\).

Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that $$\mathbf{AR} = \mathbf{B}$$ The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]
2. Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]