Finding eigenvalues for symbolic matrix with known eigenvectors

Question

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?

Answer 1

HINT

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

• $[-5\quad15\quad -8]v_1=4\lambda_1$
• $[-5\quad15\quad -8]v_2=2\lambda_2$