Derivative of the product of matrices

Question

please give me a hint for the following exercise: It is the third part of this exercise

Complementar Lie subalgebra of $\mathfrak{o}(2n)$

Let $O(n)$ be the real orthogonal group and define the map $$F:O(n)\times GL(n;\mathbb{R})\to GL(n;\mathbb{R})$$ given by $(M,N)\longmapsto MN^{-1}$. Find $D_{(I,I)}F$.

---

I see that $$F(M,N)=m(M,i(N)),$$ where $m$ is the multiplication of matrices and $i$ the invertion. Also I know that

$D_{(I,I)}m(A,B)=A+B$ and $D_Ii=-I_d$

How can I continue? Thank you

Answer 1

You have all you need to deduce from the chain rule that$$DF_{(e,e)}(M,N)=M+(-N)=M-N.$$