A nonnegative matrix A is called primitive if A(k) is positive for some integer k > 0. A generalization by Protasov and Voynov (2012) of this concept to finite sets of matrices is as follows; a set of matrices, M = {A(1), A(2), ..., A(m)} is primitive if A(i1)A(i2) ... A(ik) is positive for some indices i(1), i(2), ..., i(k). The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product. We show that while primitivity is algorithmically decidable, unless P = NP it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices. On the other hand, defining P to be the set of matrices with no zero rows or columns, we give a combinatorial proof of the Protasov-Voynov characterization (2012) of primitivity for matrices in P which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Cerny on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in P. In particular, any primitive set of n x n matrices in P has a positive product of length O(n(3)). (C) 2015 Elsevier Ltd. All rights reserved.