Let \(\mathbf{A} = \begin{pmatrix} 3 & 0 \\ 1 & 1 \end{pmatrix}\).

(a) Prove by mathematical induction that, for all positive integers \(n\),

\(2\mathbf{A}^n = \begin{pmatrix} 2 \times 3^n & 0 \\ 3^n - 1 & 2 \end{pmatrix}.\)

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