Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Let $\mathcal{A}=(A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n,\mathrm{.}\mathrm{.}\mathrm{.})$ be a finite or infinite sequence of $2\times 2$ 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 $\mathrm{\ne }2$. We also describe canonical forms for sequences of $2\times 2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.