Answer to Question #313056 in Abstract Algebra for Sophia

Question

Determine whether the given set of invertible nXn matrices with real number entries is a subgroup of GL (n, R). The nXn matrices with determinant -1 or 1

Answer 1

Solution: Let $H$ be the set of invertible $n$ x $n$ matrices whose determinant is 1 or -1. Then we show that $H$ is a subgroup of $GL(n,R)$ :

a) First we show that $H$ is closed. Suppose $A,B\in H,$ which means $det(A)\in \{1,-1\}$ and $det(B)\in\{1,-1\}$ .Then $det(AB)=det(A)det(B)$ can only be $1$ or $-1$. But this means $AB$ satisfies the requirement for being in $H$ , so $AB\in H,$ hence $H$ is closed.

b) The identity $I$ is in $H$ because $det(I)=1,$ meaning $I$ meets the requirement for being in $H$.

c) Suppose $A\in H$ . This means $det(A)$ is either $1$ or $-1$ . Hence $det(A^{-1})=\frac{1}{det (A)}$ is either $1$ or $-1$ , so $A^{-1} \in H$ .

Properties (a),(b),(c) above show that H is a subgroup of $GL(n,R).$