## Problem on nonzero solutions to a homogeneous linear system

For what value of $c$ is there a nonzero solution to the following equation? For that value of $c$, find all solutions to the equation. $$\begin{pmatrix}1&1\\2&c\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}$$
• ## Solution

#### Finding the value of $c$

Recall that
We proceed by computing the determinant:
If $c=2$, the determinant is zero, and thus there are infinitely many solutions to the equation. All but one of them are nonzero.

#### Finding all solutions for $c=2$

Plugging in $c=2$, we now try to find all solutions $\begin{pmatrix}x\\y\end{pmatrix}$ to the matrix equation $$\begin{pmatrix}1&1\\2&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}.$$
The first row of this equation reads $x+y=0$. The second row reads $2x+2y=0$.
These equations are redundant. Any $(x,y)$ such that $y=-x$ is a solution.
That is, any point of the form $(x,-x)$ is a solution.