Question: Let $f$ be a permutation of the set {$1, 2, ..., n$} (a one-to-one function from the set to itself). Let $P(f)$ be the $n \times n$ matrix whose ($i, j$)-entry is $\delta_{i, f(j)}$. Show that $P(f^{-1}) = P(f)^{-1}$.
Thoughts: This should be pretty straightforward I think—nothing overly complicated, however, clearly I'm not entirely sure as I have decided to post the question here. Does it suffice to prove that $P(f)P(f^{-1}) = I_{n}$? If so, why exactly is that the case? As usual, I know where I need to end but need help getting started.