Find Eigen values of given matrix with nonfactorable polynomial

Question

I'm having trouble finding the Eigen values for this matrix:

$$ A =\begin{pmatrix} 0&1&-2 \\ 1&3&0 \\ -2&0&5 \end{pmatrix} $$

I did $A - \lambda I $ and ended up with this matrix:

$$ A - \lambda I =\begin{pmatrix} -\lambda&1&-2 \\ 1&3-\lambda&0 \\ -2&0&5-\lambda \end{pmatrix} $$

I then took the determinant and got $ -\lambda^3 + 8 \lambda^2 - 10\lambda - 17 $, but I don't know what I can do from here. The above polynomial is not factorable. How would I find the Eigen values?

Answer 1

As @Jared mentioned. You have to solve it numerically. I used Mathematica to obtain the following numbers. $$ \lambda_1 \approx -0.929142304121661 \\ \lambda_2 \approx 3.18552786994252 \\ \lambda_3 \approx 5.74361443417914\\ $$