Are There n!/2 Even/Odd Permutation Matrices for nxn?

[Terrell]

is there a proof that the number of even/odd permutation matrices of any nxn, where n is greater than 3, is n!/2? basically, i want to understand the derivation of n!/2. thank you!

 

[mfb]

It is easy to show that (a) there are in total n! permutation matrices and (b) there is a bijective function between odd and even permutations (e.g. swap two images). Combine both and you get n!/2 odd and n!/2 even permutations.

 

[Terrell]

It is easy to show that (a) there are in total n! permutation matrices and (b) there is a bijective function between odd and even permutations (e.g. swap two images). Combine both and you get n!/2 odd and n!/2 even permutations.

how can i show the bijectiveness? sorry if my question is stupid

 

[mfb]

By constructing it. For each even permutation, find exactly one unique odd permutation, or vice versa. I gave an example how to do that.