TITLE: Is the inverse function smooth?
QUESTION [5 upvotes]: Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often differentiable?

REPLY [5 votes]: Hint: The inverse is the adjugate matrix divided by the determinant, hence its entries are ratios of two polynomials (i.e. a rational function) in the entries of $A$. So, you are essentially asking why a rational function is infinitely differentiable (provided that the denominator is nonzero).