# finding eigenvector from 3x3 matrix

I have

$$A = \begin{bmatrix} 5 & -2 & 1 \\ -2 & 2 & -2 \\ -1 & -2 & 5 \end{bmatrix}$$

which has eigenvalues $\lambda_1 = \lambda_2 = 6$ and $\lambda_3 = 0$. I want to find the eigenvectors for these eigenvalues.

I've tried to turn it into equations and trying to solve them (for $\lambda_1 = \lambda_2 = 6$) $$-x -2y - z = 0 \\ -2x - 4y - 2z = 0 \\ -x -2y - z = 0$$

But no matter how I try to approach this, I have no idea how to get my eigenvectors. I know from the answer that the eigenvector for $\lambda_1$ & $\lambda_2$ should be: