This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post.

This problem is from assignment 4.

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

A permutation matrix $P$ has a single 1 in each row and a single 1 in each column, all other entries being 0. So column $j$ has a single 1 at position $e_{i_jj}$. $P$ acts by moving row $j$ to row $i_j$ for each column $j$. Taking the transpose of $P$ moves each 1 entry from $e_{i_jj}$ to $e_{ji_j}$. Then $P^t$ acts by moving row $i_j$ to row $j$ for each row $i_j$. Since this is the inverse operation, $P^t=P^{-1}$.

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

Thanks.