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

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Statement

Let R be any commutative unital ring and GL(2,R) be the General linear group (?) of degree two over R. Then, the Transpose-inverse map (?) is a composite of division by the determinant and conjugation by the matrix \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

In other words, for any matrix A \in GL(2,R):

(A^t)^{-1} = \frac{1}{\det A} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} A \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}^{-1}.

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