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

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Statement

Suppose R is any commutative unital ring. Let G = GL(2,R) be the 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 Projective general linear group (?) of degree two over R.

Then, the automorphism of PGL(2,R) induced by the Transpose-inverse map (?) automorphism of G is an inner automorphism of PGL(2,R).

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