Let SL2 R be the set of all 2 times 2matrices a b c d with a

Let SL_2 (R) be the set of all 2 times 2-matrices [a b c d] with a, b, c, d elementof R and ad - bc = 1. Show that SL_2 (R) is a group under matrix multiplication.

Solution

for a,b,c,d belong to R det( [a,b][c,d]) = ad-bc

for n * n matrices det(AB) = detA * detB = 1 * 1 = 1

so, AB belongs to SL(2,R)

hence it is closed under multiplication

Associativity follows from the associativity of matirx multiplication

also I ( [1,0] [0,1]) belongs to SL(2.R) and A . I = I .A = I

hence SL(2,R) has identity element

A = ( [a,b] [c,d])

A-1 = ( [d,-b] [-c,a]) and belongs to SL(2.R)

hence for every element inverse exists

hence SL(2,R) is group under matrix multiplication