1. Prove that the set of 2 x 2 matrices with entries from Z and determinant equal to 1 is a group under matrix multiplication. 2. Let G = {[aa] R,a+ a Prove that G is a group under matrix multiplication. It's a bit weird that these elements are invertible despite having determinant 0, isn't it? Comment on this. 3. Suppose that (G, *) and (H, *) are groups. Define the direct product of G and H to be the set GXH {(g, h) | g6G,H E H} with operation : defined by (91, h) (92, h2) = (91 * 92, h * h2). a) Prove that (G x H,) is a group. b) If G and H both have finite order, what is the order of G X H? c) Let G (Z3, *) be Z3 as a multiplicative group and H (Z2, +) be Z2 as an additive group. Write out all elements of G x H and write out the full operation table for G x H