Let \(n\geq 1\). The general linear group is \[ \GL_n(\RR) \coloneqq \theset{ A \in \Mat_{n\times n}(\RR) \st \det A \neq 0 }, \] a group under matrix multiplication.
A matrix \(A\in\Mat_{n\times n}(\RR)\) lies in \(\GL_n(\RR)\) if and only if there exists \(B\in\Mat_{n\times n}(\RR)\) with \(AB=BA=I_n\). If \(AB=I_n\), then \(\det A\det B=1\), so \(\det A\neq 0\); conversely, if \(\det A\neq0\), then \(B=(\det A)^{-1}\operatorname{adj}(A)\) satisfies \(AB=BA=I_n\).