Solution
= Solution
The <transpose of a matrix product> reverses factor order:
$$
\boxed{(AB)^T=B^TA^T.}
$$
For the requested two-dimensional check, put $A=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}e&f\\g&h\end{smallmatrix}\right)$. Both sides are
$$
\begin{pmatrix}ae+bg&ce+dg\\af+bh&cf+dh\end{pmatrix}.
$$
More generally, the entry formula $(AB)_{ji}=\sum_kA_{jk}B_{ki}$ is exactly $(B^TA^T)_{ij}$.