Given a matrix with some variant entries in it or a matrix expression with some variant matrices in it, the rank of the partial matrix or matrix expression will vary with respect to the variant entries or variant matrices. Many problems in matrix theory and applications are closely related to maximal and minimal possible ranks of matrix expressions with variant matrices. In this paper, we shall find the maximal and minimal ranks of the matrix expression with respect to three variant matrices.

generalized inverse, invariance, matrix equation, matrix expression, range, rank.