Axis reflection/Angle/Eigenvectors and axes/Exercise

Let

${\displaystyle {}M={\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}\,}$

be a plane reflection at an axis, ${\displaystyle {}\alpha \neq 0}$. Show that ${\displaystyle {}{\begin{pmatrix}-\sin \alpha \\\cos \alpha -1\end{pmatrix}}}$ is an eigenvector for the eigenvalue ${\displaystyle {}1}$, and that ${\displaystyle {}{\begin{pmatrix}\cos \alpha -1\\\sin \alpha \end{pmatrix}}}$ is an eigenvector for the eigenvalue ${\displaystyle {}-1}$ of ${\displaystyle {}M}$.