The matrix A, given by

$\displaystyle A=\left(\begin{array}{ccc}7&-4&6\\2&2&2\\-3&4&-2\end{array}\right)$

has eigenvalues 1,2,4. Find the set of corresponding vectors.

Every time I perform these operations I only get my answers as 0's. I'll geve an example.

$\displaystyle x(A-\delta{I})=0$ taking $\displaystyle \delta$ as 1

$\displaystyle \left(\begin{array}{ccc}6&-4&6\\2&1&2\\-3&4&-3\end{array}\right)$$\displaystyle \left(\begin{array}{c}x\\y\\z\end{array}\right)=\l eft(\begin{array}{c}0\\0\\0\end{array}\right)$

Solving

$\displaystyle 6x-4y+6z=0$

$\displaystyle 2x+y+2z=0$

$\displaystyle -3x+4y-3z=0$

$\displaystyle -x+5y-z=0$

$\displaystyle x=5y-z$

$\displaystyle 30y-6z-4y+6z=0$

$\displaystyle y=0$

And so on. Could someone please direct me as to what I am doing wrong?