rank of partitioned matrix with column space information

Question

Let us consider a matrix $X=[A,B,C,D]$. Assume that $[A,B,C]$ has full column rank, but $D$ might be rank deficient. Also, assume that $col(A) \cap col(D) = \emptyset$, $col(B) \cap col(D) = \emptyset$, and $col(C) \cap col(D) = \emptyset$. In this case, does $rank(X) = rank[A,B,C] + rank(D)$ hold? My cocern is whether there exist some linear combination of $[A,B,C]$ such that $col[A,B,C] \cap col(D)$ is not empty...

Answer 1

Considering your $A,B,C,D$ are matrices of the same height, you don’t necessarily have $$ rank \begin{pmatrix} A&B&C&D \end{pmatrix} = rank \begin{pmatrix} A&B&C \end{pmatrix} + rank D$$ You only have $$ rank \begin{pmatrix} A&B&C&D \end{pmatrix} \leq rank \begin{pmatrix} A&B&C \end{pmatrix} + rank D$$ Since colums of $D$ can be in the generated vector space of colums of $A,B$ and $C$, (the converse is also true) an easy counterexample is given by $A+B=D$