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

==Statement==

Let <math>R</math> be any [[commutative unital ring]] and <math>GL(2,R)</math> be the [[fact about::general linear group]] of [[fact about::general linear group of degree two|degree two]] over <math>R</math>. Then, the [[fact about::transpose-inverse map]] is a composite of division by the determinant and conjugation by the matrix <math>\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}</math>.

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

<math>(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}</math>.

==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]]