6.
$$\mathbf { P } = \left( \begin{array} { c c }
- \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\end{array} \right)$$
- Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\).
The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
- Find the value of \(p\) and the value of \(q\).
The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
- Write down the matrix \(\mathbf { Q }\).
The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
- Find the matrix \(\mathbf { R }\).
- Deduce that the transformation \(T\) is self-inverse.