On unit stable range one matrices

Abstract

We characterize the unit stable range one 2x2 and 3x3 matrices over commutative rings. In particular, we characterize the 2x2 matrices which satisfy the Goodearl-Menal condition. For 2x2 integral matrices we show that the stable range one and the unit stable range one properties are equivalent, and, that the only matrix which satisfies the Goodearl-Menal condition is the zero matrix.