Are the vector expressions $\vec{x}^TA\vec{r}$ and $\vec{r}^TA \vec{x}$ equivalent?

Question

Are these vector expressions equal? $$\vec{x}^TA\vec{r} \qquad\vec{r}^TA \vec{x}$$

Can I combine them?

A is symmetric.

Answer 1

If $A$ is symmetric, then the relationship holds, since $(x^TAr)^T=r^TA^Tx=r^TAx$, and since the result is a scalar, equality holds (all scalars are symmetric). If $A$ is not symmetric, it is easy to come up with counterexamples.