- (i)
$$\mathbf { P } = \left( \begin{array} { r r r }
k & - 2 & 7
- 3 & - 5 & 2
k & k & 4
\end{array} \right)$$
where \(k\) is a constant
Show that \(\mathbf { P }\) is non-singular for all real values of \(k\).
(ii)
$$\mathbf { Q } = \left( \begin{array} { r r }
2 & - 1
- 3 & 0
\end{array} \right)$$
The matrix \(\mathbf { Q }\) represents a linear transformation \(T\)
Under \(T\), the point \(A ( a , 2 )\) and the point \(B ( 4 , - a )\), where \(a\) is a constant, are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\) respectively.
Given that the distance \(A ^ { \prime } B ^ { \prime }\) is \(\sqrt { 58 }\), determine the possible values of \(a\).