3 This question concerns the matrix \(\mathbf { M }\) where \(\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3
4 & - 3 & - 2
2 & 1 & 4 \end{array} \right)\).

1. Obtain the characteristic equation of \(\mathbf { M }\). Find the eigenvalues of \(\mathbf { M }\). These eigenvalues are denoted by \(\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }\), where \(\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }\).
2. Verify that an eigenvector corresponding to \(\lambda _ { 1 }\) is \(\left( \begin{array} { r } 1
   3
   - 1 \end{array} \right)\) and that an eigenvector corresponding to \(\lambda _ { 2 }\) is \(\left( \begin{array} { r } 1
   2
   - 1 \end{array} \right)\). Find an eigenvector of the form \(\left( \begin{array} { l } a
   1
   c \end{array} \right)\) corresponding to \(\lambda _ { 3 }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\). (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).) Hence write down an expression for \(\mathbf { M } ^ { 4 }\) in terms of \(\mathbf { P }\) and a diagonal matrix. You should give the elements of the diagonal matrix explicitly.
4. Use the Cayley-Hamilton theorem to obtain an expression for \(\mathbf { M } ^ { 4 }\) as a linear combination of \(\mathbf { M }\) and \(\mathbf { M } ^ { 2 }\).