7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0
2 & 1 \end{array} \right)\).
- Show that \(\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0
80 & 1 \end{array} \right)\). - Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\), where \(n\) is a positive integer.
- Use induction to prove that your answer to part (ii) is correct.