I have a very stupid question. Consider two permutation matrices (containing only 0s and 1s) that are equivalent. Is it true that both have equal number of 1s?
I think this should be true because you do not seem to be able to change the number of 1s by elementary row operations.
If this is the case, does this give a satisfactory proof of the parity of permutations expressed as transpositions being the same?
I feel like I'm mixing up some things, but don't exactly understand how.
If they are really permutation matrices they will have exactly one $1$ in each row/column, so the answer is yes without requiring that they are equivalent (they should have the same dimension, of course)