Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} 5-\lambda & 34 & -41 \\ 4 & 17-\lambda & -23 \\ 5 & 24 & -31-\lambda \end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \end{bmatrix}$
$\begin{bmatrix} 5-\lambda & 34 & -41 \\ 4 & 17-\lambda & -23 \\ 5 & 24 & -31-\lambda \end{bmatrix}=0$
$(\lambda+2)^2(\lambda+5)=0$
$\lambda_1=\lambda_2=-2, \lambda_3=-5$
2. Find eigenvectors:
For $\lambda=-2$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} 7 & 34 & -41 \\ 4 & 19 & -23 \\ 5 & 24 & -29 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \end{bmatrix} $
Let $r,s$ be free variables.
$\vec{V}=r(1,1,1)+s(1,1,1)\\
E_1=\{(1,1,1);(1,1,1)\} \\
\rightarrow dim(E_2)=2$
For $\lambda=-5$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} 10 & 34 & -41 \\ 4 & 22 & -23 \\ 5 & 24 & -16 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix} $
Let $t$ be a free variable.
$\vec{V}=t(20,11,14)\\
E_2=\{(20,11,14)\} \\
\rightarrow dim(E_2)=1$
