9.
$$\mathbf { A } = \left( \begin{array} { c c }
k & - 2
1 - k & k
\end{array} \right) \quad \text { where } k \text { is a constant }$$
- Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(k\).
A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
The point \(P\) has position vector \(\binom { a } { 2 a }\) relative to an origin \(O\).
The point \(Q\) has position vector \(\binom { 7 } { - 3 }\) relative to \(O\).
Given that the point \(P\) is mapped onto the point \(Q\) under \(T\), - determine the value of \(a\) and the value of \(k\).
Given that, for a different value of \(k , T\) maps the line \(y = 2 x\) onto itself,
- determine this value of \(k\).