We consider how to take X such that the linear matrix expression A - BXC attains its maximal and minimal ranks, respectively. As applications, we investigate the rank invariance and the range invariance of A - BXC with respect to the choice of X. In addition, we also give the general solution of the rank equation rank(A - BXC) + rank(BXC) = rank(A) and then determine the minimal rank of A - BXC subject to this equation.