Transpose-inverse map induces inner automorphism on projective general linear group of degree two

Statement
Suppose $$R$$ is any commutative unital ring. Let $$G = GL(2,R)$$ be the fact about::general linear group of degree two over $$R$$, $$Z$$ be the center of $$G$$ (which is also the group of scalar matrices, because Center of general linear group is group of scalar matrices over center), and let $$PGL(2,R) = G/Z$$ be the fact about::projective general linear group of degree two over $$R$$.

Then, the automorphism of $$PGL(2,R)$$ induced by the fact about::transpose-inverse map automorphism of $$G$$ is an inner automorphism of $$PGL(2,R)$$.

Related facts

 * Transpose-inverse map is inner automorphism on special linear group of degree two
 * Transpose-inverse map is composite of inner automorphism and division by determinant on general linear group of degree two