3 Let \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 5 & 2
5 & 3 & - 2
2 & - 2 & - 4 \end{array} \right)\).
- Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } - 2 \lambda ^ { 2 } - 48 \lambda = 0\).
You are given that \(\left( \begin{array} { r } 1
- 1
1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) corresponding to the eigenvalue 0 . - Find the other two eigenvalues of \(\mathbf { M }\), and corresponding eigenvectors.
- Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { P } ^ { - 1 } \mathbf { M } ^ { 2 } \mathbf { P } = \mathbf { D }\).
- Use the Cayley-Hamilton theorem to find integers \(a\) and \(b\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M }\).
Section B (18 marks)