In this paper, we discuss the existence of an algorithm to decide if a given set of $2 \times 2$ matrices is mortal: a set $F=\{A_1,\dots,A_m\}$ of $2 \times 2$ matrices is said to be \motnouv{mortal} if there exist an integer $k \ge 1$ and some integers $i_1,i_2,\dots,i_k \in \{1, \ldots, m\}$ with $A_{i_1} A_{i_2} \cdots A_{i_k}=0$. 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 left upper corner problem and to the reachability problem for piecewise affine functions. Finally, we state some NP-completeness results.