5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by
$$\mathbf { A } = \left[ \begin{array} { c c }
k & k
k & - k
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c }
- k & k
k & k
\end{array} \right]$$
where \(k\) is a constant.
- Find, in terms of \(k\) :
- \(\mathbf { A } + \mathbf { B }\);
- \(\mathbf { A } ^ { 2 }\).
- Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
- It is now given that \(k = 1\).
- Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
- The matrix \(\mathbf { A }\) 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.