The mortality problem for matrices of low dimensions

In this paper we discuss the existence of an algorithm to decide if a given set of 2 × 2 matrices is mortal. A set F = {A₁,..., A_m} of square matrices is said to be moral if there exist an integer k ≥ 1 and some integers i₁, i₂,..., i_k ≥ {1,..., m} with A_i1 A_i2 ⋯ A_ik = 0. We survey this problem and propose some new extensions. We prove the problem to be BSS-undecidable for real matrices and Turing-decidable for two rational matrices. We relate the problem for rational matrices to the entry-equivalence problem, to the zero-in-the-corner problem, and to the reachability problem for piecewise-affine functions. Finally, we state some NP-completeness results.