I do this exercise but I don't know if this okay. $$P^{-1}_{\sigma}=P^{T}_{\sigma}=P_{\sigma^{-1}}$$ Where $P_{\sigma}=[e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]$ The first thing that I do is proving that $$P^{-1}_{\sigma}=P^{T}_{\sigma}$$ And we got $$P^{-1}_{\sigma}=[e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]^{-1}=e_{\sigma(1)}^{-1},e_{\sigma(2)}^{-1},...,e_{\sigma(n)}^{-1}= e_{\sigma(n)},e_{\sigma(n-1)},...,e_{\sigma(1)}= [e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]^{T}=P^{T}_{\sigma}$$ Now we prove that $$P^{-1}_{\sigma}=P_{\sigma^{-1}}$$ So we got $$ P^{-1}_{\sigma}=[e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]^{-1}=e_{\sigma(1)}^{-1},e_{\sigma(2)}^{-1},...,e_{\sigma(n)}^{-1}=e_{\sigma(1)^{-1}},e_{\sigma(2)^{-1}},...,e_{\sigma(n)^{-1}}=P_{\sigma^{-1}} $$ And after that we got $$P^{-1}_{\sigma}=P^{T}_{\sigma}=P_{\sigma^{-1}}$$

This is my solution and I want to know if is right. :)