Matsumoto’s Theorem is part of classical algebraic K-theory.There are two types of proof.Both are quite technical; a lot of matrix computation is needed.One (Matsumoto’s) is based on the Bruhat decompostion of matrices over a field,the other (my proof) uses Bass’ computation of relative reciprocities on a Dedekind domain together with reciprocity in the Milnor ring.With higher Ktheory as starting point the proof can be short.The main effort then lies in setting up K-theory and the identification of its lower part with the classical theory.