Are there any real values of $c$ for which there is a nontrivial (nonzero) solution to $$\begin{pmatrix}1&c\\-c&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}?$$
For any real value of $c$, $2+c^2$ is positive, meaning the determinant is nonzero. Hence for any value of $c$, there is only one solution to the matrix equation, the trivial solution $\begin{pmatrix}0\\0\end{pmatrix}$.