3.
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & k & - 2
2 & - 4 & 1
1 & 2 & 3
\end{array} \right)$$
where \(k\) is a constant.
- Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by
$$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$
Given that \(\operatorname { det } \mathbf { M } = 5\)
- find the value of \(k\)
- use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).