A \(3 \times 3\) matrix \(\mathbf { M }\) has characteristic equation
$$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$
Show that \(\lambda = 2\) is an eigenvalue of \(\mathbf { M }\). Find the other eigenvalues.
An eigenvector corresponding to \(\lambda = 2\) is \(\left( \begin{array} { r } 3 - 3 1 \end{array} \right)\).
Evaluate \(\mathbf { M } \left( \begin{array} { r } 3 - 3 1 \end{array} \right)\) and \(\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1 - 1 \frac { 1 } { 3 } \end{array} \right)\).
Solve the equation \(\mathbf { M } \left( \begin{array} { l } x y z \end{array} \right) = \left( \begin{array} { r } 3 - 3 1 \end{array} \right)\).
Find constants \(A , B , C\) such that
$$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
A \(2 \times 2\) matrix \(\mathbf { N }\) has eigenvalues -1 and 2, with eigenvectors \(\binom { 1 } { 2 }\) and \(\binom { - 1 } { 1 }\) respectively. Find \(\mathbf { N }\).
Section B (18 marks)