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

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- Solution i

Solution

The transpose reverses the product order, so the definition is

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

If

A

and

B

are orthogonal matrices, then

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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