Eigenvectors must stay independent

Let $M$ be a $3\times 3$ real matrix. Let $\begin{pmatrix}1\\2\\3\end{pmatrix}$, $\begin{pmatrix}1\\1\\1\end{pmatrix}$ and $\begin{pmatrix}0\\-1\\\alpha\end{pmatrix}$ be the eigenvectors of $M$ corresponding to three distinct eigenvalues of $M$, where $\alpha$ is a real number. Then, which of the following is NOT a possible value of $\alpha$?