 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/25j/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The differential of determinant at

A

D (det)_{A} (X) = det (A) tr (A^{- 1} X) .

It is nonzero at every

A \in S L (n, R)

, so the regular-level-set theorem makes

S L (n, R)

n^{2} - 1

. At the identity,

T_{I} S L (n, R) = {X : tr X = 0} .

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