 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/3/b/iii/solution

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

Write the general real symmetric matrix as

E_{2} = (r t t u)

. With the specified

E_{1}

, anticommutation reads

E_{1} E_{2} + E_{2} E_{1} = (2 t r + u r + u 2 t) = 0.

Hence

t = 0

and

u = - r

. The remaining condition

E_{2}^{2} = I

gives

r^{2} = 1

, leaving exactly

E_{2} = \pm (10 0 - 1) .

Both matrices directly satisfy all the conditions.

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