Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} 7-\lambda & -2 & -2\\ 0 & 4-\lambda & -1\\
-1 & 1 & 4-\lambda \end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix}$
$\begin{bmatrix} 7-\lambda & -2 & -2\\ 0 & 4-\lambda & -1\\
-1 & 1 & 4-\lambda \end{bmatrix}=0$
$-( \lambda-5)^3=0$
$\lambda_1=\lambda_2=\lambda_3=5$
2. Find eigenvectors:
For $\lambda=5$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} 7-\lambda & -2 & -2\\ 0 & 4-\lambda & -1\\
-1 & 1 & 4-\lambda \end{bmatrix}=\begin{bmatrix} 2 & -2 & -2\\ 0 & -1 & -1\\ -1 & 1 & -1 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix} $
Let $r,s,t$ be free variables.
$\vec{V}=r(2,1,1)+s(2,0,1)+t(1,0,0)\\
E_1=\{(2,1,1);(2,0,1);(1,0,0)\} \\
\rightarrow dim(E_2)=3$
Hence, $S=\begin{bmatrix} 2& 2 & 1\\ 1 & 0 & 0 \\
-1 & 1 & 0 \end{bmatrix}$
