Transpose-inverse map induces class-inverting automorphism on projective general linear group

Statement
Suppose $$F$$ is a field and $$n$$ is a natural number. Let $$G = GL(n,F)$$ be the general linear group and $$P = PGL(n,F)$$ be the fact about::projective general linear group, i.e., the quotient group $$G/Z(G)$$. Then, the transpose-inverse map on $$G$$ induces an automorphism on $$P$$ which is a  class-inverting automorphism.

Related facts

 * Transpose-inverse map is class-inverting automorphism for general linear group
 * General linear group implies every element is automorphic to its inverse
 * Projective general linear group implies every element is automorphic to its inverse
 * Special linear group implies every element is automorphic to its inverse
 * Projective special linear group implies every element is automorphic to its inverse

Facts used

 * 1) uses::Transpose-inverse map is class-inverting automorphism for general linear group
 * 2) uses::Class-inverting automorphism induces class-inverting automorphism on any quotient

Proof
The proof follows directly from facts (1) and (2).