4 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
k & 0 & 2
0 & - 1 & - 1
1 & 1 & - k
\end{array} \right)$$
where \(k\) is a real constant.
- Show that \(\mathbf { A }\) is non-singular.
The matrices \(\mathbf { B }\) and \(\mathbf { C }\) are given by
$$\mathbf { B } = \left( \begin{array} { r r }
0 & - 3
- 1 & 3
0 & 0
\end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r }
- 3 & - 1 & 1
1 & 1 & 2
\end{array} \right)$$
It is given that \(\mathbf { C A B } = \left( \begin{array} { l l } - 2 & - \frac { 3 } { 2 }
- 1 & - \frac { 3 } { 2 } \end{array} \right)\). - Find the value of \(k\).
- Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).