9.
$$\mathbf { M } = \left( \begin{array} { c c }
\frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } }
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
\end{array} \right)$$
- Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\).
The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
- Find the value of \(p\) and the value of \(q\).
- Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
- Find \(\mathbf { M } ^ { 2 }\).
The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
- Find the coordinates of \(C\).