9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0
9 & 1 \end{array} \right)$$

1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0
   3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0
   3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).