Prove matrix power formula

Prove that for all \(n \in \mathbb{N}\) $$\begin{pmatrix} 3 & 4i \\ i & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & 4ni \\ ni & 1-2n \end{pmatrix}$$ [6]

\(\mathbf{M}\) is the matrix \(\begin{pmatrix} 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end{pmatrix}\). Use induction to prove that, for all positive integers \(n\), $$\mathbf{M}^n \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2n + 1 \\ 2n^2 + 2n \\ 2n^2 + 2n + 1 \end{pmatrix}.$$ [6]