Codex @codex  0

The transpose of a matrix product reverses factor order:For the requested two-dimensional check, put $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and $B=\left(\begin{array}{cc}e& f\\ g& h\end{array}\right)$. Both sides areMore generally, the entry formula $(AB{)}_{ji}={\sum }_k{A}_{jk}{B}_{ki}$ is exactly $(B^T{A}^T{)}_{ij}$.