Let B(n,k) be the set of all (0,1)-matrices of order n with constant line sum k and let be the minimum rank over B(n,k). It is known that , where is the rank of a recursively defined matrix . Brualdi, Manber and Ross showed that if and only if k|n. In this paper, we show that if and only if (n,k) satisfies one of the following three relations: (i) n≡±1 (mod k), k=2 or 3; (ii) n=k+1, k≥2; (iii) n=4q+3, k=4 and q≥1. Moreover, we obtain the exact values of for all n≥4 and determine all the possible ranks of regular (0,1)-matrices in B(n,4). We also present some positive integer pairs (n,k) such that , which gives a positive answer to a question posed by Pullman and Stanford.