 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/12/a/iii/solution

Use the order-reversal rule for the transpose of a product:

(A B)^{T} (A B) = B^{T} A^{T} A B = B^{T} I B = B^{T} B = I .

Therefore the product of two orthogonal matrices is orthogonal. The argument does not require

A

and

B

to commute.

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