- Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { l l }
a & 0
1 & b
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
a ^ { n } & 0
\frac { a ^ { n } - b ^ { n } } { a - b } & b ^ { n }
\end{array} \right)$$
where \(a\) and \(b\) are constants and \(a \neq b\).