## Problem on nontrivial solutions to a linear system

Are there any real values of $c$ for which there is a nontrivial (nonzero) solution to $$\begin{pmatrix}1&c\\-c&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}?$$
• ## Solution

Recall that
Because we are seeking values of $c$ for which there are nontrivial solutions, we require that $\text{det } \bfA = 0$.
We identify $$\bfA = \begin{pmatrix}1&c\\-c&2 \end{pmatrix}.$$
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 $\begin{pmatrix}0\\0\end{pmatrix}$.