Linear algebra. Question about symmetric matrix Prove that if

$(Au)\cdot v = u\cdot (Av)$ for every $u$ and $v$

then $A$ is symmetric.

I tried using a $2\times2$, but didn't help.
Look at $ u,v$ being standard unit vectors $e_i, e_j$. Then, what are the quantities in question?
Look at $ u,v$ being standard unit vectors $e_i, e_j$. Then, what are the quantities in question?