To differentiate $f:GL_n(\mathbb R) \to GL_n(\mathbb R)$ given by $f(A)=A^{-1}$

Question

I want to show that $f:GL_n(\mathbb R) \to GL_n(\mathbb R)$ given by $f(A)=A^{-1}$ is differentiable and find its derivative map ; now I'm convinced that the derivative map $Df_A :M_n(\mathbb R) \to M_n(\mathbb R)$ looks like $Df_A(X)=A^{-1}XA^{-1}$ and as far as I got is that

$f(A+H)-f(A)-Df_A(H)=((I+A^{-1}H)^{-1}-(I+A^{-1}H) )A^{-1}$ ; now if I can show that the norm of this is less than or equal to $K||H||^2$ , where $K$ does not depend on $H$ , then we are done ; am I correct ? and if I am , then how to show this ? And is their any other way ?

Answer 1

$Df_A(X) = - A^{-1}XA^{-1}$. Namely, differentiating $I=AA^{-1}$ at $A$ in the direction $X$ gives $0 = XA^{-1} + ADf_A(X)$. Differentiability itself follows from the implicit function theorem.