  Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of
$2\times2$ matrices with entries in an integral domain. We show that, except
for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if
and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\leq
j,k\leq\infty$. We also provide a simple numerical criterion for
triangularization.
  Using constructive methods in invariant theory, we define a map (with the
minimal number of invariants) that distinguishes simultaneous similarity
classes for non-commutative sequences over a field of characteristic $\neq2$.
We also describe canonical forms for sequences of $2\times2$ matrices over
algebraically closed fields, and give a method for finding sequences with a
given set of invariants.
