Transformation of Vector in Dot Product is Interchangable?

Question

In the answer key to a practice exam part of a proof says

$$(Ax)\cdot y = x \cdot (A^T y)$$ Where $A$ is a symmetric $n\times n$ matrix and $x$ and $y$ are eigenvectors of $A$.

What property of the dot product allows you to do this?

Answer 1

The key is that by simply looking at the definition of the dot product, $(Ax)\cdot y$ is equal to the one entry in the $1\times 1$ matrix $$y^T A x$$

Now observe that a $1\times 1$ matrix is symmetric. Hence this is also equal to $$x^T A^T y$$ whose only one entry is $(A^T y)\cdot x$.