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