 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-3/6d/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

True. Use the odd-dimensional orthogonal determinant splitting:

Φ : O (3) \to SO (3), Φ (A) = (det A) A .

Scalar matrices commute with every matrix, so

Φ (A B) = Φ (A) Φ (B)

. Its output is orthogonal and

det Φ (A) = (det A)^{3} det A = (det A)^{4} = 1.

Every

B \in SO (3)

satisfies

Φ (B) = B

, proving surjectivity. The equation

Φ (A) = I

A = (det A) I

, whence

ker Φ = {I, - I}, ∣ ker Φ∣ = 2.

- I

- 1

, which is the sign needed in this construction.

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