  We compute low-dimensional K-groups of certain rings associated with the
study of the Hermite ring conjecture. This includes a monoid ring whose
low-dimensional K-groups were recently computed by Krishna and Sarwar in the
case where the base ring is a regular ring containing the rationals. We are
able to extend their result to an arbitrary regular base ring, thereby
completing an answer to a question of Gubeladze. Our computation only relies on
certain conveniently chosen analytic patching diagrams.
  These patching diagrams also allow us to investigate stably free modules
appearing in a problem posed by Murthy. They make it possible to relate
Murthy's problem to a stable question (about the relationship between
symplectic and ordinary K-theory). More precisely, we show that Murthy's
problem has a solution if the stable question has an affirmative answer, while
a negative answer to the stable question implies that there are counterexamples
to the Hermite ring conjecture.
  Since the module appearing in Murthy's problem has rank 2, the above
mentioned argument relies on some theorems about patching with 2-by-2 matrices.
These use so-called pseudoelementary 2-by-2 matrices, a notion we introduce
which makes it possible to extend certain results valid for larger matrices to
the case of 2-by-2 matrices (if we use pseudoelementary instead of elementary
matrices). For example, we prove that the analogue of Vorst's theorem about
matrices over polynomial extensions over a regular ring containing a field
holds for 2-by-2 matrices.
