Prove that a matrix is the permutation matrix of some permutation just when
1 its entries are all 0’s or 1’s and 2 it has exactly one 1 in each column and 3 it has exactly one 1 in each row.
My attempt is that I kinda of "know" this is true because of the definition of a permutation matrix but I do not know how to describe this or prove it. I have also used the search function on this site.
I know multiplying on the left by a permutation matrix gives row rearrangements to the matrix in question. And multiplying on the right gives column rearrangements. And that if $A$ and $B$ are permutation matrices, their product does the rearrangements of $B$ followed by $A$.
I would like the answer in sigma notation as well as the capital letter notation. If I could see this worked out I believe I can make progress on the rest of the problems I want to do. Right now, I am completely lost.
(I have already asked through normal channels in my university which was unbelievably unhelpful)