3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k
1 & 1 & 3
1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6
- 1 & 3 & 1
1 & - 2 & - 2 \end{array} \right)\).

1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4
   4 & - 6 & - 10
   - 1 & 2 & 2 \end{array} \right)\).
2. Verify that \(\left( \begin{array} { l } 4
   1
   1 \end{array} \right) , \left( \begin{array} { l } 2
   1
   0 \end{array} \right)\) and \(\left( \begin{array} { r } 2
   3
   - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10
   2 & - 3 & - 5
   0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4
   - 3 & 6 & 6
   1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)