1 Let $\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 1 & 1 \end{array} \right)$.\\
(a) Prove by mathematical induction that, for all positive integers $n$,

$$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l } 
2 \times 3 ^ { n } & 0 \\
3 ^ { n } - 1 & 2
\end{array} \right)$$

(b) Find, in terms of $n$, the inverse of $\mathbf { A } ^ { n }$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q1}}