Calculating Formulas for Ranks of Submatrices of the Moore-Penrose Inverse

First, we introduce a set of formulas for calculating the ranks of Submatrices E₁, E₂, E₃ and E₄ in M⁺, and give a lot of reduced forms for these formulas under some general conditions. Then, we introduce some inequalities for ranks of submatrices in M and M⁺. Finally, we give a set of calculating formulas and some inequalities for ranks of matrices C and D in the Moore-Penrose inverse (A+iB)⁺=C+iD.