What permutations of matrix entries do row and column transpositions generate? Let $M$ be a square matrix. By transposing rows and columns, can we get any permutation of the entries of $M?$ If we can't, which permutations are generated?

Preserving column and row members will be the only obstruction (i.e. if $a$ and $b$ are in the same row originally they will be in the same row after the transformation).  So there are only $(n!)^2$ possible rearrangements via row/column transpositions, out of $(n^2)!$ possible rearrangements of the $n^2$ entries.  Your permutations are elements of $S^n \times S^n$, the cartesian product of the permutation group on $n$ letters.  For $$(\sigma, \tau) \in S^n \times S^n$$ we have $$ (a_{ij}) \mapsto (a_{\sigma(i) \tau(j)}).$$
