 Solution

ID: 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 .

0, \pm i ∣ a ∣

, so

M

1, 1 \pm 2 i ∣ a ∣

det M = 1 + 4∣ a ∣^{2} .

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

 New to topics? Read the docs here!