In Example CPMS3 we found the characteristic polynomial of \begin{equation*} F= \begin{bmatrix} -13 & -8 & -4\\ 12 & 7 & 4\\ 24 & 16 & 7 \end{bmatrix} \end{equation*} to be $\charpoly{F}{x}=-(x-3)(x+1)^2$. Factored, we can find all of its roots easily, they are $x=3$ and $x=-1$. By Theorem EMRCP, $\lambda=3$ and $\lambda=-1$ are both eigenvalues of $F$, and these are the only eigenvalues of $F$. We have found them all.