3 The matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3
- 2 & - 3 & 6
2 & 2 & - 4 \end{array} \right)\).
- Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } + 6 \lambda ^ { 2 } - 9 \lambda - 14 = 0\).
- Show that - 1 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
- Find an eigenvector corresponding to the eigenvalue - 1 .
- Verify that \(\left( \begin{array} { l } 3
0
1 \end{array} \right)\) and \(\left( \begin{array} { r } 0
3
- 2 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\). - Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { M } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
- Use the Cayley-Hamilton theorem to express \(\mathbf { M } ^ { - 1 }\) in the form \(a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\).
Section B (18 marks)