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

Table of contents
- Solution b

Solution

For every invertible matrix

A

,

(r_{C} \circ f \circ l_{C}) (A) = r_{C} ((C A)^{- 1}) = A^{- 1} C^{- 1} C = A^{- 1} = f (A) .

l_{C}

and

r_{C}

are linear. Differentiate this identity at

A = I

and use the chain rule and part (a):

r_{C} (D f (C) (C h)) = - h .

Thus

D f (C) (C h) C = - h

. Given an arbitrary increment

k

h = C^{- 1} k

to obtain the derivative of matrix inversion

D f (C) (k) = - C^{- 1} k C^{- 1} .

This also proves differentiability at every

C \in V

.

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