$$\mathbf { M } = \left( \begin{array} { c c c }
2 & a & 4
1 & - 1 & - 1
- 1 & 2 & - 1
\end{array} \right)$$
where \(a\) is a constant.
- For which values of \(a\) does the matrix \(\mathbf { M }\) have an inverse?
Given that \(\mathbf { M }\) is non-singular,
- find \(\mathbf { M } ^ { - 1 }\) in terms of \(a\)
(ii) Prove by induction that for all positive integers \(n\),
$$\left( \begin{array} { l l }
3 & 0
6 & 1
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
3 ^ { n } & 0
3 \left( 3 ^ { n } - 1 \right) & 1
\end{array} \right)$$