 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/5/i/solution

M M^{- 1} = I

and

M^{- 1} M = I

reverses the product order and gives

(M^{- 1})^{†} M^{†} = I, M^{†} (M^{- 1})^{†} = I .

Thus

(M^{- 1})^{†}

is the two-sided matrix inverse of

M^{†}

, proving

(M^{†})^{- 1} = (M^{- 1})^{†} .

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