Rank Relations Between A {0,1}-Matrix And Its Complement

Let A be a {0, 1}-matrix and r(A) denotes its rank. The complement matrix of A is defined and denoted by A(c) = J-A, where J is the matrix with each entry being 1. In particular, when A is a square {0, 1}-matrix with each diagonal entry being 0, another kind of complement matrix of A is defined and denoted by (A) over bar = J-I-A, where I is the identity matrix. We determine the possible values of r(A) +/- r(A(c)) and r(A) +/- r((A) over bar) in the general case and in the symmetric case. Our proof is constructive.