<s>[INST] Find the eigen values and vectors of the matrix Find the eigenvalues and eigenvectors of the matrix $A$:
$$A = \begin{bmatrix}-2 & 2 & -3\\2 & 1 & -6\\-1 & -2 & 0\\\end{bmatrix}.$$
$$A - \lambda I = \begin{bmatrix}-2-\lambda & 2 & -3\\2 & 1-\lambda & -6\\-1 & -2 & -\lambda\\\end{bmatrix}\\
\det(A-\lambda I)=(-2-\lambda)[(-1-\lambda)(-\lambda)-12]-2(-2\lambda-6)-3(-4-(-1+\lambda))\\
=(-2-\lambda)[(-\lambda+\lambda^2)-12]+4\lambda+12+9+3\lambda\\
=2\lambda-2\lambda^2+\lambda^2-\lambda^3+24+12\lambda+4\lambda+12+9+3\lambda\\
=-\lambda^3-\lambda^2+21\lambda+45$$
After factoring....... $\lambda=5,\lambda=-3$
When $\lambda=5$,
$$\begin{bmatrix}-7 & 2 & -3\\2 & -4 & -6\\-1 & -2 & -5\\\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$
$$-7x+2y-3z=0,\\
2x-4y-6z=0,\\
-x-2y-5z=0.$$
I am stuck here, i have no idea what to do next. I hope someone can help 
Please and thanks
 [/INST] This is easier to proceed if we permute the matrix:
$$
\pmatrix{2& -7& -3\\
-4& 2& -6\\
-2&-1&-5} \rightarrow \pmatrix{2& -7& -3\\
0& -12& -12\\
0&-8&-8}
$$
This clearly leads to the equation:
$$
-12x - 12z = 0 \rightarrow z = -x \\
\text{using the first equation, }2y - 7x - 3z = 0 \rightarrow y = \frac{7x + 3z}{2} = \frac{7x - 3x}{2} = 2x
$$
This gives the eigenvector as $\langle x, 2x, -x\rangle \leadsto \langle1, 2, -1\rangle$, or as a unit vector: $\left\langle \pm\sqrt{\frac{1}{6}}, \pm \sqrt{\frac{2}{3}}, \mp \sqrt{\frac{1}{6}}\right\rangle$
</s>