I'm note entirely sure what you mean. However, define to be the center of your group (Z stands for Zentrum; Germany was the world's mathematical superpower before the 1930, when group theory was in its infancy). This set is the set of all elements which commute with every element in your group. It is a subgroup of (in fact, it is normal too).
In the center consists of all diagonal matrices where the elements on the diagonal are constant (e.g. they are all 3, or are all 5, or are all 23514). Thus, the set of matrices which commute with every other matrix is the set,
I hope that answers the middle bit...
What do you mean by a total matrix?