7. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\)
Prove by induction that \(\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\) for all integers \(n \geq 1\)
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