 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/2/c/solution

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

True. Set

C_{ij} = A_{ki} B_{k ℓ} A_{ℓ j}

. Exchange

i, j

, interchange the dummy indices

k, ℓ

, and use

B_{ℓ k} = B_{k ℓ}

C_{ji} = A_{kj} B_{k ℓ} A_{ℓ i} = A_{ℓ j} B_{ℓ k} A_{ki} = A_{ki} B_{k ℓ} A_{ℓ j} = C_{ij} .

A^{T} B A

preserves symmetry, even when

A

is singular.

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