rank of partitioned matrix with rank equality in sub martrices

Question

Let us consider a matrix $X=[A,B,C]$. Assume that $rank[A,B]=rank(A)+rank(B)$, $rank[A,C]=rank(A)+rank(C)$ and $rank[B,C]=rank(B)+rank(C)$.

Then, does this imply $rank[A,B,C]=rank(A)+rank(B)+rank(C)$ or $rank[A,B,C]=rank[A,B]+rank(C)$? If true, how can I prove it?

Answer 1

This does not work in general. Consider $A=[1,1]^T$, $B=[1, 0]^T$, $C=[0, 1]^T$. Then $$[A, B, C] = \left[\begin{matrix}1 &1 & 0 \\ 1 &0 & 1\end{matrix}\right]$$ but $rank([A,B,C]) = 2\neq rank(A) + rank(B)+ rank(C) = 3$. Also $rank([A,B,C])\neq rank([A,B]) + rank(C) = 3$