Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 6 / c / ii

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- Solution ii

Solution

The contraction is

tr (A^{3})

. Since

A^{T} = - A

, transpose and invariance of the trace give

tr (A^{3}) = tr ((A^{3})^{T}) = tr ((- A)^{3}) = - tr (A^{3}) = 0.

This is the odd-power case of vanishing odd complete contractions of an antisymmetric matrix.

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