Prove that the product of two matrices in SL2R has determina

Prove that the product of two matrices in SL_2(R) has determinant one.

Solution

SL₂(R) = {set of all 2x2 matrices over R whose determinet is equal to 1}

Let A, B are in SL₂(R)

This implies det(A) = 1 = det(B)

Now consider det(AB) = det(A) x det(B)

=1x1 =1

Therefore the product of two matrices in SL₂(R) has determinant one