Find the missing values of this 3x3 matrix A, $$A=\begin{pmatrix}-3 & a & b\\\ c & 1 & d \\\ e & f & -1\end{pmatrix}$$ given the eigenpairs $$λ_1 = -2, x_1 = \begin{pmatrix}-1 \\\ 1 \\\ 1 \end{pmatrix}$$ and, $$λ_2 = 1, x_2 = \begin{pmatrix}1 \\\ 0 \\\ 1 \end{pmatrix}$$
I have been going through a couple eigenvalue and eigenvector problems in my textbook and found this one particularly interesting, although I'm not quite sure how to approach this problem. I have tried looking for similar problems in other books, however, I was not able to find one that would give me a general idea.
I think it's possible the question has a typo? By definition, if $(\lambda, v)$ is an eigenvalue/eigenvector pair, then $Av = \lambda v.$ If you do this for $(\lambda_2, x_2),$ then you get $$\begin{bmatrix} b+1 \\ c+d \\ e-4 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \\ -1 \end{bmatrix},$$ which implies that $c+d=0.$ But if you do this for the first eigenpair, then you get $$\begin{bmatrix} 1+2a+b \\ c+4+d \\ e+2f-4 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix},$$ and comparing the second entries yields $c+4+d = 2 \Rightarrow c+d = -2,$ which seems to yield a contradiction.