Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$.
I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong M_{n \times n}(\mathbb{R})$.
I am confused about what atlas to use to prove this. My thoughts were to find a set $U \subset \mathbb{R}^{n^2}$ such that $\ f:U \to \mathrm{GL}(n,\mathbb{R})$ is given by $f(x) = x$ for all $x \in U$.
If I can somehow define the set $U$ so that it maps homeomorphically to $\mathrm{GL}(n,\mathbb{R})$ through the identity map, then I would be done. But this hinges on the fact that $U$ should be an open set, which I am unsure how to prove.
Through some reading online I also saw that the cases $\mathrm{det}(A) >0 $ and $\mathrm{det}(A)<0$ should be considered separately. Can someone push me in a starting direction?
Hint the determinant is continuous so the inverse image of the real line -0 by the determinant is an open subset so it is a submanifold.