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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?

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

• $[-5quad15quad -8]v_1=4lambda_1$


• $[-5quad15quad -8]v_2=2lambda_2$