 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/d/solution

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

True. A symmetric tensor and an antisymmetric tensor are orthogonal under full index contraction. Renaming the two summed indices gives

A_{ij} B_{ij} = A_{ji} B_{ji} = (- A_{ij}) B_{ij} = - A_{ij} B_{ij},

A_{ij} B_{ij} = 0

. This is a scalar statement with both indices summed, not a claim that every individual entrywise product vanishes.

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