Keele Research Repository

In this paper, we consider decidability questions that are related to the membership problem in matrix semigroups. In particular, we consider the membership of a given invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimension 4 with integral matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.