Inversion is a homeomorphism of GL_n

Idea. The formula $A^{-1}=\frac{1}{\det A}\operatorname{adj}A$ expresses the inverse as a polynomial map divided by a nonvanishing polynomial.

Step 1: continuity. On $GL_n(\mathbb{R})$ the function $\det$ is continuous and nonzero (Problem 66), so $A\mapsto\frac1{\det A}$ is continuous as the composition of $\det$ with $t\mapsto1/t$ on $\mathbb{R}\setminus\{0\}$. The map $A\mapsto\operatorname{adj}A$ is continuous (polynomial entries). The product of a continuous scalar function with a continuous matrix-valued map is continuous. Hence $\iota:A\mapsto A^{-1}$ is continuous.

Step 2: homeomorphism. $GL_n(\mathbb{R})$ is closed under inversion, so $\iota:GL_n\to GL_n$ is well defined. Moreover $\iota(\iota(A))=(A^{-1})^{-1}=A$, so $\iota\circ\iota=\mathrm{id}$. Thus $\iota$ is a continuous bijection whose inverse is itself, hence continuous: $\iota$ is a homeomorphism (an *involution*).

Example. For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$, visibly continuous where $ad-bc\ne0$.

Remark.
- Inversion is not defined on all of $M_n$, and $\|A^{-1}\|\to\infty$ as $A$ approaches a singular matrix (Problem 64 in the previous part). So it does not extend continuously to the closure.
- Inversion exchanges the two connected components of $GL_n$ only in the sense that it preserves each: $\det(A^{-1})=1/\det A$ has the same sign.