simple symmetric matrix identity

Question

Is this true if $\mathbf{A}$ is symmetric? $$ \frac{1}{2}e^\intercal\mathbf{A}x+\frac{1}{2}x^\intercal\mathbf{A}e = e^\intercal \mathbf{A}x$$ ($e$ and $x$ are vectors.)

Update

Is this proof right?

$e, x$ are vectors. so $e^\intercal \mathbf{A}x$ is scalar.
$k = k^\intercal$ if $k$ is scalar.
so $e^\intercal \mathbf{A}x = x^\intercal \mathbf{A}^\intercal e = x^\intercal \mathbf{A} e$. then we can rewrite the question as a form of $\frac{1}{2}a + \frac{1}{2}a = a$

I hope this is right.

Answer 1

Yes, your proof is correct. $ $