Problem

(i) Prove by induction that, for $n \in \mathbb{Z}^+$

$\begin{pmatrix} -2 & 9 \\ -1 & 4 \end{pmatrix}^n = \begin{pmatrix} 1 - 3n & 9n \\ -n & 3n + 1 \end{pmatrix}$

(ii) A sequence of numbers is defined by

$u_1 = 1 \quad u_2 = 4 \quad u_{n+2} = 6u_{n+1} - 9u_n$

Prove by induction that, for $n \in \mathbb{Z}^+$

$u_n = 3^{n-2}(n + 2)$