When is a pair of matrices mortal?

Abstract

A set of matrices over the integers is said to be k-mortal (with k positive integer) if the zero matrix can be expressed as a product of length k of matrices in the set. The set is said to be mortal if it is k-mortal for some finite k. We show that the problem of deciding whether a pair of 48 x 48 integer matrices is mortal is undecidable, and that the problem of deciding, for a given k, whether a pair of matrices is k-mortal is NP-complete. © 1997 Elsevier Science B.V.