Let D be the unit disk. Kutzschebauch and Studer [8] recently proved that, for each continuous map A : D → SL(2, C), which is holomorphic in D, there exist continuous maps E, F : D → sl(2, C), which are holomorphic in D, such that A = e E e F . Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.MSC 2020: 47A56, 15A54, 15A16, 30H50. Keywords: holomorphic matrices, bordered Riemann surfaces, exponentials. 1 In the sense of [1, II.3A], which includes that X is connected. For example, X can be the closure of a bounded smooth domain X in the complex plane C.2 For X = D, this is well-known [7]. I do not know if this is already known for non-simply connected X.