Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 11E / b / Solution

For every invertible matrix $A$,The maps $l_C$ and $r_C$ are linear. Differentiate this identity at $A=I$ and use the chain rule and part (a):Thus $Df(C)(Ch)C=-h$. Given an arbitrary increment $k$, set $h=C^{-1}k$ to obtain the derivative of matrix inversionThis also proves differentiability at every $C\in V$.