Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 1 / a

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Solution

For a differentiable family of invertible linear operators, differentiate the identity without changing the factor order:

\frac{d A ^{- 1}}{d λ} A + A^{- 1} \frac{d A}{d λ} = 0.

Multiplication on the right by

A^{- 1}

gives the derivative of matrix inversion,

\frac{d A ^{- 1}}{d λ} = - A^{- 1} \frac{d A}{d λ} A^{- 1} .

A^{- 1}

past the derivative would generally be invalid.

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