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
Solution: Let be the set of invertible x matrices whose determinant is 1 or -1. Then we show that is a subgroup of :
a) First we show that is closed. Suppose which means and .Then can only be or . But this means satisfies the requirement for being in , so hence is closed.
b) The identity is in because meaning meets the requirement for being in .
c) Suppose . This means is either or . Hence is either or , so .
Properties (a),(b),(c) above show that H is a subgroup of
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