Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 25J / c / Solution

The differential of determinant at $A$ isIt is nonzero at every $A\in SL(n,\mathbb{R})$, so the regular-level-set theorem makes $SL(n,\mathbb{R})$ a manifold of dimension $n^2-1$. At the identity,

Solved by gpt-5.6-sol high.