On unit stable range matrices

Abstract

We characterize the unit stable range one \(2\times 2\) and \(3\times 3\) matrices over commutative rings. In particular, we characterize the \(2\times 2\) matrices which satisfy the Goodearl-Menal condition. For \(2\times 2\) 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.