Intuitive explanation for inverse of a permutation matrix Today in lecture we learned that the transpose of a permutation matrix is the inverse of the permutation matrix. Meaning,
$$P^{T}P = I$$
I can work out the math by matrix multiplication but I'd prefer a deeper, more intuitive understanding.
What I have so far in my head is:
We know that the matrix $P$ will swap rows when we apply it to a matrix, let's say $A$. Then $PA$ will swap the $i^{th}$ row of A with the $j^{th}$ of $A$.
This then means that $P^{T}(PA)$ must swap our new $i^{th}$ row with the new $j^{th}$ row so we can have our original $A$ matrix back. Why is this always true? More specifically why does $P^{T}$ swap back out rows...?

A permutation matrix $P$ preserves length. So $\|Px\|=\|x\|$, which gives $\langle P^TPx,x\rangle=\langle x,x\rangle$ in the Euclidean inner product. That's enough to imply that $P^TP=I$. So, $P^T$ has to undo any swapping done by $P$.
