Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} 1-\lambda & -3 & 1 \\
-1 & -1-\lambda & 1 \\ -1 & -3 & 3-\lambda
\end{bmatrix}\begin{bmatrix}
v_1\\
v_2 \\
v_3
\end{bmatrix}=\begin{bmatrix}
0\\
0 \\
0
\end{bmatrix}$
$\begin{bmatrix} 1-\lambda & -3 & 1 \\
-1 & -1-\lambda & 1 \\ -1 & -3 & 3-\lambda
\end{bmatrix}=0$
$-( \lambda -2)^2(\lambda +1)=0$
$\lambda_1=2 + 3i, \lambda_2=-2-3i$
$\dim =1 \ne 2$
Hence, matrix $A$ is non-defective.
