Nonnegative primitive matrices with exponent 2

A nonnegative matrix M with zero trace is primitive if for some positive integer k, Mk is positive. The exponent exp(M) of the primitive matrix is the smallest such k. By treating the digraph G whose adjacency matrix is the primitive matrix M, we will show that the minimum number of positive entries of M is 3n − 3 when exp(M) = 2. We will also show that for a symmetric n × n matrix M if exp(M) = 2, the minimum number of positive entries of M is 3n − 2 or 3n − 3 according to n.