What is the relation between ranks of these two matrices

Question

I have matrices A and B. Using those matrices I calculate matrix C, as: $$ C = I - A\,(A'B^{-1} A)\,AB^{-1} $$ The matrix C doesn’t have full rank, i.e. it is singular matrix with $\text{rank}(C)=K$. Now, I take submatrices of A and B, by selecting rows and columns related to correlated variables, and I know in advance which rows and columns needs to be selected. Let’s have notation: D is submatrix of A and E is submatrix of B. Using this submatrices, I calculate new matrix $$ F = I - D\,(D'E^{-1}D)\,DE^{-1} $$ The question is: is there any rule/relationship between $\text{rank}(C)$ and $\text{rank}(F)$ ? Is it possible that $\text{rank}(F)>K$ ?