Problem

$$M = \begin{pmatrix} 1 & -1 & 4 \\ 3 & a & b \\ a & 1 & b \end{pmatrix}$$

where $a$ and $b$ are constants.

Given that $\mathbf{i} + 2\mathbf{j} + \mathbf{k}$ is an eigenvector of $M$,

(a) determine the corresponding eigenvalue.

(b) Hence determine the value of $a$ and the value of $b$.

(c) Determine

(i) the other eigenvalues of $M$,

(ii) eigenvectors which correspond to these eigenvalues.