Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
a reflection in the line \(y = - x\);
a stretch parallel to the \(y\)-axis of scale factor 7 .
Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = - x\) followed by a stretch parallel to the \(y\)-axis of scale factor 7 .
The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 3 & - \sqrt { 3 } - \sqrt { 3 } & 3 \end{array} \right]\).
Show that \(\mathbf { A } ^ { 2 } = k \mathbf { I }\), where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
Show that the matrix \(\mathbf { A }\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = ( \tan \theta ) x\). [0pt]
[5 marks]