TITLE: If $f(A)\to A^{-1}$, prove that $f$ is continuous.
QUESTION [10 upvotes]: Let $f \colon GL_{n}(\mathbb{R})\to GL_{n}(\mathbb{R})$ be a function which maps $A\mapsto A^{-1}$. Prove that $f$ is continuous.
$GL_{n}(\mathbb{R})=\det^{-1}(\mathbb{R}\setminus\{0\})$ is the set of invertible matrix of  order $n\times n$ with real coefficients.



Should I do $A^{-1}=\dfrac{\operatorname{adj}(A)}{\det(A)}$ and use it like a rational polynomial?
Are there a overkill way to prove this? Any hint? 
Thanks MSE!

REPLY [2 votes]: An easy way to see that inversion is continuous (and in fact smooth) is to use the matrix inversion algorithm by elementary row operations that you probably learned in your first linear algebra course: Gauss-Jordan elimination. Using this algorithm, we obtain $A^{-1}$ from $A$ by matrix multiplication by elementary matrices. Since matrix multiplication is smooth, and the composite of smooth maps is smooth, then in fact the inversion is smooth.