Anticommutativity in the ring of square matrices of the second order with complex entries

The main purpose of this paper is to solve the equation AB+BA =0 in the ringM2(C) of square matrices of the second order with complex entries. The discussion is made by considering the cases when A and B are inversable or singular. The methods used in each case are completely different and instructive. Considerations about matrices which commutes and ?nally an application are also given.