Assume your matrix group is over the field K. Then, the center consists of all scalar matrices whose determinant is 1 or -1.

Similarly, the center consists of all scalar matrices whose determinant is 1.what about the center when the determinant of the elements in the group all have det=1?

In this case, the center is isomorphic to the group of n-th roots of unity in K.

Edit: In this case, just n=2.