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

Statement
Let $$R$$ be any commutative unital ring and $$GL(2,R)$$ be the fact about::general linear group of degree two over $$R$$. Then, the fact about::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 $$A \in GL(2,R)$$:

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

Related facts

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