Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k 1 & 0 & 1 3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 1 & - 8 & 2 1 & - 5 & 1 \end{array} \right)\).
The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 1 3 \end{array} \right) , \left( \begin{array} { r } - 1 0 1 \end{array} \right)\) and \(\left( \begin{array} { l } 4 1 2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\).
Section B (18 marks)