When do we have $(AB)^T= AB$?

Question

When is it true that $(AB)^T= AB$? I'd say that if $A$ commutes with $B$ it is true, but i don't know if there is any other case where this is true.

Answer 1

If $A$ and $B$ are symmetric, then $(AB)^T=AB$ iff $B^TA^T=BA=AB$, that is your relation is true iff $A$ and $B$ commute.

However, if $A$ or $B$ is not symmetric, $A$ and $B$ need not commute, here is a counterexample:

$$A=\begin{bmatrix}1&3\\2&1\end{bmatrix}$$ $$B=\begin{bmatrix}6&1\\1&4\end{bmatrix}$$

Then $(AB)^T=AB$ but $AB\ne BA$.