I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) .
The 1.1.3 (b) is :
Let F: $U\rightarrow U$ be defined as $A\rightarrow A^{-1}$. Show that $% D_{A}F(H)=-A^{-1}HA^{-1}$ for any $n\times n$ matrix $H.$
$D_{A}F(H)$ is the Frechet derivative of F at A. H I guess should be the small "drifting", so that it is actually trying to calculate the derivative of $(A+tH)^{-1}$
I tried to expand $(A+H)^{-1}$ as $(I+A^{-1}H)^{-1}A^{-1}=I-A^{-1}H+\frac{% A^{-2}H^{2}}{2!}-\frac{A^{-3}H^{3}}{3!}+...,$ but it doesn't looks like $% -A^{-1}HA^{-1}$
Some one can give me a hint?