I am looking for two matrices $A$ and $B$ such that:
$$A\begin{bmatrix} a & b & c \\ d & e & f \\g & h & i \\j & k & l \\m & n & o \\p & q & r \\\end{bmatrix}B = \begin{bmatrix}a & d & b & e & c & f\\ g & j & h & k & i & l \\ m & p & n & q & o & r\end{bmatrix}$$
Can anyone tell me whether such matrices exist and, if so, give their values and, if not, disprove that such matrices can exist?
The answer is no; no such matrices exist.
One proof is as follows: suppose for contradiction that such an $A,B$ exist. Then if we take $a=j=n=1$ and set all other entries equal to zero, then we find that $$ A \pmatrix{ 1\\ \\ \\ 1\\ &1&\\&} B=\pmatrix{1\\&1\\&&1&&&}. $$ Note that the matrix on the right has rank $3$. However, for any $M$, we have $\operatorname{rank}(AMB) \leq \operatorname{rank}(M)$.