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Math and science::Algebra::Aluffi

Subgroups of the general linear group

Below are three subgroups of \( \mathrm{GL}_n(\mathbb{R})\), the group of invertible \( n\times n\) matrices with real entries.

\( \mathrm{SL}_n(\mathbb{R})\)
The subset of \( \mathrm{GL}_n(\mathbb{R})\) consisting of [what matricies?]. "S" is short for "special".
\( \mathrm{O}_n(\mathbb{R})\)
The subset of \( \mathrm{GL}_n(\mathbb{R})\) consisting of [what matricies?]. "O" here is short for "orthogonal".
\( \mathrm{SO}_n(\mathbb{R})\)
The subset of \( \mathrm{O}_n(\mathbb{R})\) consisting of [what matricies?].

Can you proof that these 3 subsets form subgroups?