= Orientation-preserving general linear group
{title2=$\operatorname{GL}^+(n,\mathbb R)=\{A:\det A>0\}$}
The real invertible matrices with positive <determinant> form a <subgroup> of the <general linear group>. The identity has determinant one, <multiplicativity of the determinant> preserves positivity under products, and $\det(A^{-1})=1/\det A>0$ preserves it under inversion. More precisely this is the <kernel of a group homomorphism> $A\mapsto\operatorname{sgn}(\det A)$ to $\{1,-1\}$, so it is a <normal subgroup> of index two for $n\geq1$. Its elements preserve an <orientation of a vector space>.
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