Question #313056

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

1
Expert's answer
2022-03-18T09:30:21-0400

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

a) First we show that HH is closed. Suppose A,BH,A,B\in H, which means det(A){1,1}det(A)\in \{1,-1\} and det(B){1,1}det(B)\in\{1,-1\} .Then det(AB)=det(A)det(B)det(AB)=det(A)det(B) can only be 11 or 1-1. But this means ABAB satisfies the requirement for being in HH , so ABH,AB\in H, hence HH is closed.

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

c) Suppose AHA\in H . This means det(A)det(A) is either 11 or 1-1 . Hence det(A1)=1det(A)det(A^{-1})=\frac{1}{det (A)} is either 11 or 1-1 , so A1HA^{-1} \in H .

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


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