# Finding eigenvalues for symbolic matrix with known eigenvectors

I could not find anything specifically like this using search, so my problem is:

Find eigenvalues $$\lambda_1$$ and $$\lambda_2$$ for matrix $$A$$ when we know two of its eigenvectors. Solve variables $$a$$, $$b$$, $$c$$ and $$d$$.

$$A = \left[ \begin{matrix} a & b & 10 \\ c & d & 0 \\ -5 & 15 & -8 \end{matrix} \right] , v_1 = \left[ \begin{matrix} 1 \\ 3 \\ 4 \end{matrix} \right] , v_2 = \left[ \begin{matrix} -1 \\ 1 \\ 2 \end{matrix} \right]$$

So if we go straight ahead and try to solve eigenvalues by characteristic polynomial, we end up with a massive polynomial with five variables and no solutions. How should I start solving this as the usual algorithm (find characteristic polynomial -> solve $$\lambda$$ -> solve rref($$A-\lambda I_n$$) -> determine variables) is no use?

For a full solutions, let consider and solve the system of $$6$$ equations in $$6$$ unknowns
• $$Av_1=\lambda_1v_1$$
• $$Av_2=\lambda_2v_2$$
If we need only to find the eigenvalues let consider only the third row of $$A$$ to obtain
• $$[-5\quad15\quad -8]v_1=4\lambda_1$$
• $$[-5\quad15\quad -8]v_2=2\lambda_2$$