6 The matrix \(\mathbf { M }\) is defined by
$$\mathbf { M } = \left[ \begin{array} { c c }
\sqrt { 3 } & 3
3 & - \sqrt { 3 }
\end{array} \right]$$
- Show that
$$\mathbf { M } ^ { 2 } = p \mathbf { I }$$
where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
- Show that the matrix \(\mathbf { M }\) can be written in the form
$$q \left[ \begin{array} { c c }
\cos 60 ^ { \circ } & \sin 60 ^ { \circ }
\sin 60 ^ { \circ } & - \cos 60 ^ { \circ }
\end{array} \right]$$
where \(q\) is a real number. Give the value of \(q\) in surd form.
- The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection.
Find:
- the scale factor of the enlargement;
- the equation of the mirror line of the reflection.
- Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).