Past exam of the short options course of the University of Oxford / 2014 / 4 / b

Factor $A+H=A(I+A^{-1}H)$. The determinant expansion near the identity isthe linear terms in the permutation formula come from choosing one diagonal increment and all remaining identity entries. ThereforeThe required Fréchet derivative is the linear functionalThe quadratic remainder proves full differentiability, rather than only the existence of each directional derivative. This is the derivative of the determinant.