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