Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix}
5-\lambda & 0\\
0 & 5-\lambda
\end{bmatrix}\begin{bmatrix}
v_1\\
v_2
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$
$\begin{bmatrix}
5-\lambda & 0\\
0 & 5-\lambda
\end{bmatrix}=0$
$(\lambda -5) ^2=0$
$\lambda_1=\lambda_2=5$
2. Find eigenvectors:
For $\lambda=5$
let $B=A-\lambda_1I$
$B=\begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix}$
Let $r$ be a free variable.
$\vec{V}=r(1,0) +s(0,1) \\
E_1=\{(1,0);(0,1)\}
\rightarrow dim(E_1)=2$
Hence, $A$ is non-defective.
