I was trying to solve a coupled torsion pendulum (note this is for an assignment, so please just guidance, no solutions), consisting of two rods suspended on a vertical wire clamped at either end. I've obtained the matrix equation $ \begin{bmatrix} -\Omega & \Omega\\ \Omega & -\frac{1}{3}\Omega\\ \end{bmatrix} \begin{bmatrix} A\\ B\\ \end{bmatrix} = \lambda \begin{bmatrix} A\\ B\\ \end{bmatrix} $
From here I formed the characteristic equation $\lambda^2 + \frac{4}{3}\Omega\lambda - \frac{2}{3}\Omega^2 = 0$, and found the eigenvalues $\lambda = -\frac{1}{3}\Omega(2 + \sqrt{10})$ and $\frac{1}{3}\Omega(\sqrt{10} - 2)$ (which I checked with WolframAlpha).
Then, something strange happened... when I tried to use these eigenvalues to compute the eigenvectors (both by hand and with WolframAlpha), I found I couldn't. The resulting matrices had non-zero determinants (interestingly when I first put the matrix into WolframAlpha, it gave me the eigenvalues and eigenvectors, but when I plugged those eigenvalues back in, it did not give me eigenvectors).
My question is, have I done something fundamentally wrong, or made a simple mistake? I have been over it myself but I can't find any simple errors, which leads me to believe that perhaps my approach is wrong. Honestly I'm just stumped that the eigenvalues aren't giving me eigenvectors, it doesn't seem possible.