<s>[INST] Eigenvectors - unknown variables Considering that $v_1$ and $v_2$ are eigenvectors, find the values of $a,b,c,d$
:
$$
A = \begin{bmatrix} 1 & a & b \\1 & c & d\\
1 & 1 & 1\end{bmatrix}\qquad v_1 = \begin{bmatrix} 1 \\1 \\
1 \end{bmatrix}\qquad v_2 = \begin{bmatrix} 1 \\0 \\
-1 \end{bmatrix}
$$
I am honestly quite confused as to how I can solve this. I tried to find det but I found this:
$c-d-a(1-d)+b(1-c)$
which is not very useful
 [/INST] $Av_1=\lambda_1v_1\Rightarrow \begin{pmatrix}1+a+b\\1+c+d\\3\end{pmatrix}=\lambda_1\begin{pmatrix}1\\1\\1\end{pmatrix}$ and $Av_2=\lambda_2v_2\Rightarrow \begin{pmatrix}1-b\\1-d\\0\end{pmatrix}=\lambda_2\begin{pmatrix}1\\0\\-1\end{pmatrix}$
Then $\lambda_2=0, b,d=1$. Now we can find $a \text{ and }c$ too.
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