7.
$$\mathbf { P } = \left( \begin{array} { c c }
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
\end{array} \right)$$
- Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\).
The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the \(x\)-axis.
- Write down the matrix \(\mathbf { Q }\).
Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
- find the matrix \(\mathbf { R }\).
- Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.