 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/5/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Since

A

and

B

U^{†} = A - i B

. Expanding the unitarity condition gives

I = U^{†} U = (A - i B) (A + i B) = A^{2} + B^{2} + i (A B - B A) .

Equality of the entrywise real and imaginary parts yields

A^{2} + B^{2} = I, A B = B A .

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