Let's say I have these 2 matrices:
$$A = \begin{bmatrix} c \\ d \end{bmatrix} $$ and
$$B = \begin{bmatrix} e \\ f \end{bmatrix}$$
$A'B = ce + df $ and $B'A = ec + fd$
As shown above, $A'B = B'A$. But is there a matrix product property that could've told me this without having to manually check? I haven't been able to find any matrix properties that prove that $A'B = B'A$
A real number can be considered a $1\times 1$ real matrix, which is trivially symmetric. Therefore if $A^TB = C\in\mathbb{R}$, then $A^TB = C = C^T = B^TA$.