Past exam of the short options course of the University of Oxford / 2014 / 4 / d / Solution

Apply the regular level set theorem to $F(A)=\det A-1$ on the $n^2$-dimensional vector space of real matrices. At a point with determinant one, part (b) gives $D{F}_A(H)=\mathrm{tr}(A^{-1}H)$. Choosing $H=A/n$ gives $D{F}_A(H)=1$, so this derivative is surjective.

Hence the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth submanifold of dimension $n^2-1$. At the identity,This is the special linear Lie algebra. When $n=1$, the group is a single point and the tangent space is zero, consistent with the same formula.