Consider an $n \times p, (n>p)$ matrix$X=[A,B,C]$ where $A$ is $n \times p_A$, $B$ is $n \times p_B$ and $C$ is $n \times p_C$ with $n=n_A+n_B+n_C$. Assume that submatrices $[A,B]$, $[B,C]$ and $[A,C]$ are all of full column rank. In this case, can we say that $X$ is of full column rank?
Or more generally, for a matrix $Z=[Z_1,...Z_m]$, can we say that $Z$ is of full column rank if $[Z_i,Z_j]$ are of full column rank for all $i \neq j$?
No it’s not true. The problem is that columns of $C$ can be linear combination of columns of $A$ and $B$. An example is
$$X:=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\\ \end{pmatrix}$$
and let $A:= \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}$, $B:= \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \\ \end{pmatrix}$, $C:= \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\ \end{pmatrix}$.