Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 5 / a

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- Solution a

Solution

The inverse is assumed to exist. First,

(I + M) (I - M) = I - M^{2} = (I - M) (I + M) .

By matrix commutation with an inverse, multiplying on both sides by

(I - M)^{- 1}

gives

(I + M) (I - M)^{- 1} = (I - M)^{- 1} (I + M) .

An equivalent calculation writes

I + M = 2 I - (I - M)

, so both products equal

2 (I - M)^{- 1} - I

. No convergent infinite matrix series is needed.

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