Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 7 / b / ii / Solution

The antisymmetric contribution is the negative of twice the cross-product matrix for

a

:

M = 1 - 2 a_{3} 2 a_{2} 2 a_{3} 1 - 2 a_{1} - 2 a_{2} 2 a_{1} 1 .

The cross-product matrix spectrum is

0, \pm i ∣ a ∣

, so

M

has eigenvalues

1, 1 \pm 2 i ∣ a ∣

. Multiplication gives

det M = 1 + 4∣ a ∣^{2} .

Direct expansion of the displayed matrix yields the same real positive determinant.

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