 Solution

ID: 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/solution

The inverse is assumed to exist. First,

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

(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.

 New to topics? Read the docs here!