Question

Define a permutation matrix and prove that its transpose is its inverse.

Answer 1

Prove that the transpose of a permutation matrix PP is its inverse.

A permutation matrix PP has a single 1 in each row and a single 1 in each column, all other entries being 0. So column jj has a single 1 at position eijjeijj. PP acts by moving row jj to row ijij for each column jj. Taking the transpose of PP moves each 1 entry from eijjeijj to ejijejij. Then PtPt acts by moving row ijij to row jj for each row ijij. Since this is the inverse operation, Pt=P−1Pt=P−1.

Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.

Thanks.