Rank of matrix - with more information (python)

I'm working on a specific class of n-square matrices where the rank is n when n is even, and n-1 when n is odd and I would like to understand why.

numpy.linalg.rank was useful to notice this fact; however, to understand why it happens I would like to have a non-zero linear combination of my lines - or columns - that sum to all-zeros.

I checked how numpy.linalg.rank works and it seems that it won't work (or I did not understood what I read here)

I have no idea how do to that. Is this question too specific so that no existing function does that?

source https://stackoverflow.com/questions/71465038/rank-of-matrix-with-more-information-python