Ok, where should you start? What have you done?
If you expand the determinant on the middle row you get which obviously has as a root.
Since the expansion you give obviously has 11 as a root, try factoring that quadratic as so that 6 is a double eigenvalue and 11 is an eigenvalue. Now start looking for eigenvectors.
(Double click on the formulas to see the LaTex code.)
that's a 3x3 matrix, just don't know how to put it all in one.
for \lamda =11
\left|\begin{array}{ccc}1 & 0 & -2 \\ 0 & 5 & 0\\ 2 & 0 & 4\end{array}\right|
y=0
x=2z eigenvector \left|\begin{array}{ccc}2 & 0 & 1\end{array}\right|
for \lamda=6
\left|\begin{array}{ccc}-4 & 0 & -2 \\ 0 & 0 & 0\\ -2 & 0 & -1\end{array}\right|
y=0
x=-2z eigenvector \left|\begin{array}{ccc}1 & 0 & -2\end{array}\right|
that look ok? so then i just make them orthonormal and i'm done?