Prove A is symmetric matrix iff $A$ is square and $x^T Ay = (Ax)^T y$ Prove A is a symmetric matrix iff $A$ is square and $x^T Ay = (Ax)^T y$. (for all $x,y \in \mathbb{R}^n$)
Going from the assumption that it is symmetric to the two conditions is fairly straightforward.
However, going the other way, I am stuck at proving $A^T = A$ using the second and condition, being stuck at $X^T (A-A^T)y=0$.
Note T is for transpose!

First, lets remember the rules of transposing the product: $(AB)^T = B^T A^T$
Using this, we start with the giver equation $ x^TAy=(Ax)^Ty$ and apply the rule above which yields 
$x^TAy = x^TA^Ty$.
Note that the equation is true for all $x, y$, so the only way that is possible is if $A = A^T$.
