Let An be the set of n × n zero-one matrices satisfying the matrix equation A ² = Jn, where Jn is n × n matrices of all ones. In this article, it is proved that the number of non-isomorphic left loops of order k gives the lower bound to the size of An for n = k ² . Mainly we have established that any matrix in An corresponding to loop has rank 2k − 2, where n = k ², for some positive integer k.