 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/3/solution

Q_{1}, \dots, Q_{m}

of the same size, transpose reverses the order of multiplication. Therefore

(Q_{1} \dots Q_{m})^{T} (Q_{1} \dots Q_{m}) = Q_{m}^{T} \dots Q_{1}^{T} Q_{1} \dots Q_{m} = I,

by canceling the adjacent pairs

Q_{j}^{T} Q_{j} = I

successively. Hence every finite product is an orthogonal matrix.

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