Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} 4-\lambda & 1 & 1 \\ 1 & 4-\lambda & 1\\ 1 & 1 & 4-\lambda \end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \end{bmatrix}$
$\begin{bmatrix} 4-\lambda & 1 & 1 \\ 1 & 4-\lambda & 1\\ 1 & 1 & 4-\lambda \end{bmatrix}=0$
$(\lambda-3)^2(\lambda-6)=0$
$\lambda_1=\lambda_2=3, \lambda_3=6$
2. Find eigenvectors:
For $\lambda=3$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} 4-\lambda & 1 & 1 \\ 1 & 4-\lambda & 1\\ 1 & 1 & 4-\lambda \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \end{bmatrix} $
Let $r,s$ be free variables.
$\vec{V}=r(1,0,-1)+s(0,1,-1)\\
E_1=\{(1,0,-1);(0,1,-1)\} \\
\rightarrow dim(E_2)=2$
For $\lambda=6$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix} $
Let $t$ be a free variable.
$\vec{V}=t(1,1,1)\\
E_2=\{(1,1,1)\} \\
\rightarrow dim(E_2)=1$
