Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$$M = \begin{pmatrix} 4 & 0 & 2 \\ 0 & 4 & a \\ 2 & a & \frac{20}{3} \end{pmatrix}$$

where $a$ is a constant.

Given that $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ is an eigenvector of M,

(a) determine the corresponding eigenvalue for this eigenvector.

(b) Hence show that $a = 4$

Determine

(c) the other two eigenvalues of M,

(d) a normalised eigenvector for each eigenvalue of M.

(e) Hence determine a matrix P and a diagonal matrix D such that

$$P^TMP = D$$