The simple fact that the invertible square matrices of order $n$ are precisely those which have nonzero determinant has a real bunch of theoretical applications (in ring theory, topology, differential geometry, etc).

For example, if you consider the general linear group $GL(n, \mathbb{R})$ as a Lie group, to see that its associated Lie algebra is (isomorphic to) $gl(n, \mathbb{R})$, an essential step is to get the equivalence of tangent planes $T_e(GL(n,\mathbb{R})) \cong T_e(gl(n,\mathbb{R}))$ (where $e$ is the neutral element). But this is trivial if we have in mind that $GL(n,\mathbb{R}) = \{A\in gl(n,\mathbb{R}) : det(A) \neq 0 \}$ and that $det$ is continuous, because then $GL(n,\mathbb{R})$ is automatically an open set of $gl(n,\mathbb{R})$.