Using surd forms where appropriate, find the matrix which represents:
a rotation about the origin through \(30 ^ { \circ }\) anticlockwise;
a reflection in the line \(y = \frac { 1 } { \sqrt { 3 } } x\).
The matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left[ \begin{array} { c c }
1 & \sqrt { 3 }
\sqrt { 3 } & - 1
\end{array} \right]$$
represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
The transformation represented by \(\mathbf { A }\) is followed by the transformation represented by \(\mathbf { B }\), where
$$\mathbf { B } = \left[ \begin{array} { c c }
\sqrt { 3 } & - 1
1 & \sqrt { 3 }
\end{array} \right]$$
Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.