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Abstract Algebra

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Subgroups of the Invertible Matrices

ABSALG-MEMMMX

Let $G$ be a group with neutral element $e$.

A subgroup $H$ of $G$ is a subset of $G$ that is closed under group composition and under taking inverses.

Let ${\rm GL}(n,{\mathbb R})$ be the group of real $n$ by $n$ matrices with non-zero determinant, $n\ge 1$.

It is called the General Linear Group.

Which of the following subsets of ${\rm GL}(n,{\mathbb R})$ are NOT subgroups of ${\rm GL}(n,{\mathbb R})$?

Select ALL that apply.

A

The matrices with determinant equal $1$

B

The matrices with determinant equal $-1$

C

The upper triangular matrices with non-zero determinant

D

The diagonal matrices with non-zero determinant

E

The diagonal matrices with diagonal entries in $\{1,2,1/2\}$

F

The matrices with non-zero determinant that are either upper triangular or are lower triangular

G

The matrices with determinant equal $2$