Problem

$$A=\begin{pmatrix} a&a&1\\ -a&4&0\\ 4&a&5 \end{pmatrix}$$

where $a$ is a positive constant

Given that $2$ is an eigenvalue of $A$

(a) determine

(i) the value of $a$

(ii) the other two eigenvalues of $A$

A normalised eigenvector for the eigenvalue $2$ is

$$\begin{pmatrix} \frac1{\sqrt6}\\ \frac1{\sqrt6}\\ -\frac2{\sqrt6} \end{pmatrix}$$

(b) Determine a normalised eigenvector for each of the other eigenvalues of $A$

(c) Hence determine a matrix $P$ and a diagonal matrix $D$ such that $P^TAP=D$