I have a square matrix $A$ which is $m\times m$ large, for example: $$ A=\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix} $$ and a shuffled version of it, for example: $$ B=\begin{pmatrix} a & e & b\\ d & c & i\\ h & g & f \end{pmatrix} $$ Is it possible to prove that $B$ is a product of $A$ undergoing an unknown number of row and column exchanges? (for example, swap row 1 with row 2, then, swap column 2 with column 3, ...)
As far as I know, I can disprove it if they have different values for the absolute value of their determinant, but say they have the same determinant or one has the negative determinant of the other?
How about if the matrix is not square?