Let's consider the $r \times c$ matrix: \begin{equation*} \left[ \begin{array}{ccc} A & {B} & {C}% \end{array}% \right] =\left[ \begin{array}{ccc} {A}_{1} & {B}_{1} & {0} \\ {A}_{2} & {B}_{2} & {C}_{2}% \end{array}% \right] \end{equation*} where $A_1,(r_1 \times c_1)$ has full column rank, $B_1,(r_1 \times c_2)$ has full column rank and $A_1 x \neq B_1 y$ for any $x \neq 0$ and $y \neq 0$. Assume that the number of rows of $[A, B]$ is larger than that of columns, i.e., $r>c_1+c_2$, but $r_1 \ge c_1+c_2$ does not necesarily hold.
Then, (i) I conjecture that $rank[A,B,C] = rank[A,B]+rank[C]$ holds since any linear combination of $A_1$, $B_1$ and $[A_1, B_1]$ cannot be a zero vector. (ii) I conjecture that $[A,B]$ has full column rank since $A_1 x \neq B_1 y$ for any $x \neq 0$ and $y \neq 0$ and $r >c$.
I'm wondering if my above conjectures are correct. Any comments are appreciated! Thanks!