Clean integral 2 × 2 matrices are characterized. Up to similarity the strongly clean matrices are completely determined and large classes of uniquely clean matrices are found. In particular, classes of uniquely clean matrices which are not strongly clean are found.
Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.
Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.