6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9
1 & - 2 \end{array} \right)\).
- Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n
n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\). - A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively.
Determine whether the student is correct.