Show that the characteristic equation of the matrix
$$\mathbf { M } = \left( \begin{array} { r r r }
3 & - 1 & 2
- 4 & 3 & 2
2 & 1 & - 1
\end{array} \right)$$
is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } - 7 \lambda + 35 = 0\).
Show that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and find its other eigenvalues.
Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 5\).
State the magnitudes and directions of the vectors \(\mathbf { M } ^ { 2 } \mathbf { v }\) and \(\mathbf { M } ^ { - 1 } \mathbf { v }\).
Use the Cayley-Hamilton theorem to find the constants \(a , b , c\) such that
$$\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I } .$$
Section B (18 marks)