Let $a,b,c,d \in \mathbb{C}$ satisfy $ad-bc \ne 0$, and let $T: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ be the [Möbius transformation](/page/M%C3%B6bius%20Transformation) represented by these entries:
with the standard extended-value interpretation on $\widehat{\mathbb{C}}$. Then $T$ is bijective, and its inverse $T^{-1}: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ is represented by the entries $d,-b,-c,a$. Equivalently,