 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/17/a/i/solution

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

A

and

B

A^{T} A = A A^{T} = I

and

B^{T} B = B B^{T} = I

. Hence

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

Thus $C$ is orthogonal. The original PDF includes the transpose of the entire product

A B

, which is essential to the expression used here.

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