Problem

The matrix $M$ is given by

$$M = \begin{pmatrix} 3 & -4 & k \\ 1 & -2 & k \\ 1 & -5 & 5 \end{pmatrix}$$

where $k$ is a constant.

Given that $3$ is an eigenvalue of $M$,

(a) find the value of $k$.

(b) Hence find the other two eigenvalues of $M$.

(c) Find a normalised eigenvector corresponding to the eigenvalue $3$.