Lower bounds for the rank of a matrix with zeros and ones outside the leading diagonal

We have found a lower bound on the rank of a square matrix, where every entry in the leading diagonal is neither zero nor one, but every entry outside the leading diagonal is either zero or one. The rank of such a matrix is at least half the order of the matrix. Under an additional condition, the lower bound is one higher. This condition means that some auxiliary system of linear equations has no binary solution. Examples are given showing the achievability of the lower bound. This lower bound on the rank allows us to reduce the problem of finding a binary solution to a system of linear equations, where the number of linearly independent equations is sufficiently large, to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each of which differs from binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a system of a large set of linear algebraic equations. Estimates of the running time for calculating the rank of a matrix with the SymPy computer algebra system are also given. It is shown that the matrix rank over the field of residues modulo a prime number is calculated in less time than is usually required to calculate the rank of a matrix of the same order over the field of rational numbers.