On nilpotent matrices of order 2

Sa？etak In this paper, we give several characterizations of complex nilpotent matrices of order 2, which are based on the Cayley–Hamilton theorem. In terms of traces and determinants of matrices, we describe the conditions on \[ $A,B\in\mathbb{M}_2(\mathbb{C})$ \] under which the commutator AB ？ BA is a nilpotent matrix