3.
$$\mathbf { A } = \left( \begin{array} { c r }
k & - 2
1 - k & k
\end{array} \right) , \text { where } k \text { is constant. }$$
A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
- Find the value of \(k\) for which the line \(y = 2 x\) is mapped onto itself under \(T\).
- Show that \(\mathbf { A }\) is non-singular for all values of \(k\).
- Find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
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