Transpose-inverse map is inner automorphism on special linear group of degree two

Statement

Let $R$ be any commutative unital ring and $SL(2,R)$ be the special linear group of degree two over $R$. Then, the transpose-inverse map, restricted to $SL(2,R)$, is an inner automorphism, and equals conjugation by the matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

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

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

Related facts

• Transpose-inverse map is composite of inner automorphism and division by determinant on general linear group of degree two
• Transpose-inverse map induces inner automorphism on projective general linear group of degree two
• Special unitary group of degree two equals special linear group of degree two over a finite field