Given an $n \times n$ square matrix $M$ in which every row and column contains exactly one $-1$ and one $1$, and is $0$ otherwise, I'm trying to calculate whether it's always possible to reach $-M$ through successive swapping of rows/columns.
I've worked out that it's at least possible sometimes, simply by trial-and-erroring arbitrary matrices by hand, but of course this is a long way from a proof, and it only proves that it's possible for those specific matrices. I'm not too sure where to start with this.
One thing I noticed is that if $\det(M) \neq 0$ then the number of swaps must match the evenness of $n$, because every swap will flip the sign of the determinant, and then we also have
However I'm not sure if this is relevant at all, it's just the only thing I've managed to deduce so far.