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
$$\det(-M)=(-1)^n \det(M)$$
However I'm not sure if this is relevant at all, it's just the only thing I've managed to deduce so far.
Any pointers?