I'm going to call the matrix you gave M. If we want to determine what the elements that commute with M look like in , we take an arbitrary matrix
then compute AM and MA. Since we want A to commute with M we set up the equality AM=MA. This should give a=d and c=-b. Now replace d with a and c with -b in A and we'll have what an element of the centralizer of M looks like. The last thing we need to do is make sure that the matrix A we have just found really does belong to . This means we want A to be invertible. Looking at A for a moment, we see that this is the case when or
Does this help? Let me know if anything is unclear.