Solution of this system of linear equations What are the solutions of the following system of linear equations for $a\in\mathbb{R} $?

$a^2x+2ay=3a$

$2ax+a^2y=-3a$
Hint

If $a=0$, the system becomes $0=0$ and admits an infinite amount of solutions. Assuming $a \ne 0$ we can divide both sides by $a$ to get
$$
\begin{pmatrix}a &amp; 2 \\ 2 &amp; a \end{pmatrix}
\begin{pmatrix}x \\ y \end{pmatrix}
= \begin{pmatrix}3 \\ -3 \end{pmatrix}
$$
so now apply Gaussian elimination. Can you finish the problem? You should see immediately that when $a=2$, the equations on the LHS become the same...
If $a=0$ infinitely many solutions. If $a=-2$, the same thing happening. If $a \ne 0$ and $a \ne -2$, we can get $x=-y$ and then we can solve easily.