Question: Example. Show That The Following Set Is A Subgroup Of GL(2, R): SL(2, 1) = {A E GL(2,R) | Det A = 1 = Suppose A, B E SL(2, R).

Example. Show that the following set is a subgroup of GL(2, R): SL(2,R) = {A GL(2,R) | det A = | = 1} Suppose A, B = SL(2, R). Then Hence, AB E SL(2, R). det(AB)=(det A) (det B) = 1 1 = 1. Since det I = 1, the identity matrix is in SL(2, R). Finally, if AE SL(2, R), then AA- = I implies that (det A)(det A) = det I = 1. But det A = 1, so det A-1 = 1, and hence A- SL(2, R). Therefore, SL(2, R) is a subgroup of GL(2, R).