finding the center?

Question

Find the center of GL(2,R) and justify your findings.
GL(2,R) is the group of 2×2 invertible matrices of real numbers.

Answer 1

Since you know elements of the center commute with *everything*, try a couple of possibilities. Suppose [a b // c d] is in the center. Multiplying that by [1 0 // 1 1] on each side and equating them shows you that b=0 and a=d. Multiplying by [ 1 1 // 0 1] on each side and equating them shows you that c=0 and a=d. So all you're left with is multiples of the identity, which obviously work.