Uniquely clean 2 × 2 invertible integral matrices - ANDRICA, Dorin; CĂLUGĂREANU , Grigore (Studia Universitatis Babeș-Bolyai Mathematica, 2017)
While units in any unital ring are strongly clean by definition, which units are uniquely clean, is a far from being simple question, even in particular rings. In this paper, the question is solved for 2 × 2 integral matrices. It turns out that uniquely clean invertible matrices are scarce: only the matrices similar I 1 0 to 0 −1l. The study is splitted into three cases: the elliptic, the parabolic and the hyperbolic cases, according to the discriminant of their characteristic polynomial. In the first two cases, units are not uniquely clean. Mathematics Subject Classification (2010): 15B36, 16U99, 11D09, 11D45.