Consider the matrix $T=\left[\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right]$

a$)$

$($a$)$ Determine the characteristic equation of $T.$

$[5$ Marks$]$

$($b$)$ Determine the eigenvalues of $T$ and the corresponding eigenvectors of $T.$

$[9$ Marks$]$

$[1$ Mark$]$

$($c$)$ Determine a basis for the eigenspace of $T.$