I'm working on an exercise problem that is as follows. Find all real values of λ for which the system has a non trivial solution

$x\_1$ + $2x\_2$ = $λx\_1$

$2x\_1$ + $x\_2$ = $λx\_2$

I'm not sure if I did it correctly but I rewrote the equation as:

$$
 \begin{matrix}
 1 & 2\\
 2 & 1\\
 \end{matrix}
$$

multiplied by 

$$
 \begin{matrix}
 x\_1\\
 x\_2\\
 \end{matrix}
$$

which equals 

$$
 \begin{matrix}
 λx\_1\\
 λx\_2\\
 \end{matrix}
$$

I then multiplied the both sides by the inverse of the the 2x2 matrix giving me just 
$$
 \begin{matrix}
 x\_1\\
 x\_2\\
 \end{matrix}
$$

on the left hand side and I solved for $x\_1$ and $x\_2$ in terms of $λx\_1$ and $λx\_2$. After this I plugged the values I got for $x\_1$ and $x\_2$ back into the first equation in the linear system. I feel like I'm on the wrong track, so any help on how to solve this would be appreciated.