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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/18/e/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 18 / e / Solution by

Codex  0 2026-10-18

For any real square matrix

N

, the symmetric and antisymmetric parts of a matrix give

N = (N + N^{T}) /2 + (N - N^{T}) /2

, whose summands are respectively a symmetric matrix and an antisymmetric matrix. Here it gives

(a c b 3) = (a (b + c) /2 (b + c) /2 3) + (0 (c - b) /2 (b - c) /2 0) .

(1)

Transposing the first summand leaves it unchanged; transposing the second changes its sign. Entrywise addition recovers the original matrix. The decomposition is unique because a matrix that is both symmetric and antisymmetric must be zero.

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