I need to find the eigenvalues/eigenvectors of the following matrix:
$A= \begin{bmatrix}-26 & 20\\-30 & 24\end{bmatrix}$
I first used the equation for eigenvectors, and sorted out the determinant:
$P_A(x) = \det (xI - A)$ $= \det\begin{bmatrix}x & 0\\0 & x\end{bmatrix} - \begin{bmatrix}-26 & 20\\-30 & 24\end{bmatrix}$ $ = \det\begin{bmatrix}x + 20 & -20\\ 30 & x - 24\end{bmatrix}$ $= (x + 20)(x - 24) - (-600)$ $= x^2 - 4x + 120$
The problem at this point, is that the trinomial $ x^2 - 4x + 120$ can't be factored into $(x + a)(x + b)$. At least I don't think. The computer on Wolfram hasn't been able to do it, so I don't think it's factorable, which would mean I can't find the two x values that would give me the eigenvalues.
So how can I get the x values from a trinomial that isn't factoable? How else could I find the matrix's eigenvalues?