Is matrix $A^TA$ always symmetric?
Ideally we've already proved both $(A^T)^T=A$ and $(AB)^T=B^T A^T$. If not, prove these first. Then $(A^T A)^T=A^T (A^T)^T =A^TA$.
We know $(AB)^T=B^TA^T$, so $(A^TA)^T=A^T(A^T)^T=A^TA$ and hence $A^TA$ is always symmetric.