# Showing inversion function for invertible matrices is differentiable

Consider the inversion function $$f:GL_n( \mathbb{R}) \rightarrow GL_n (\mathbb{R})$$ , $$f(X)=X^{-1}.$$ Where $$GL_n( \mathbb{R})$$ denotes the set of invertible $$n \times n$$ matrices over the reals.

The question wants me to show that it is a differentiable function and then to calculate its derivative. It says to think of the set as a subset of $$\mathbb{R} ^{n^{2}}$$.

I know that if the partials exist and are continuous then it is differentiable, I can't calculate the partials explicitly though since it seems too difficult, just thinking about it I know if I were to change 1 entry in the matrix keeping all others constant (this is how I interpret partial derivative of this function, is this correct?), I could find a neighbourhood around that entry such that the matrix is still invertible (since $$det:\mathbb{R}^{n \times n} \rightarrow \mathbb{R}$$ is continuous? - this has been shown in my lecture notes) Is this the correct way to go about it? I have no solutions available to me so just seeking some clarification on here to make sure my understanding isn't completely all wrong, thanks!

$$Gl(n,\mathbb{R})$$ is an open subspace of th vector space $$M(n,\mathbb{R})$$, the inverse $$X\rightarrow X^{-1}$$ is a rational function of its coordinates (expressed with the cofactor matrices) so it is differentiable.
You have $$(X+h)^{-1}=X^{-1}(I+hX^{-1})^{-1}$$ write $$hX^{-1}=u$$ with $$\|u\|<1$$, you obtain that $$(I+u)^{-1}=\sum(-1)^nu^n$$, this implies that t$$(X+h)^{-1}=X^{-1}-X^{-1}hX^{-1}+O(h)$$ and the differential is $$h\rightarrow X^{-1}hX^{-1}$$.
• Thanks for your answer! So I'm not sure what a rational function is, don't think we've actually defined that in class. I see how you've manipulated the increments to get what you got, and surely from what you end up with, the derivative is $h \rightarrow -X^{-1}hX^{-1}$ ? (minus sign) – Displayname Mar 7 at 10:11
• Also, how did you know how to go about this question i.e use that power series representation of $(I+u)^{-1}$? Is it just experience in dealing with these types of problems? – Displayname Mar 7 at 10:13