Implications between $A$, $B$, $[A,B]^T$ having full column rank where $A$ and $B$ have less columns than rows

Question

Let $A$ and $B$ be matrices having more rows than columns. It's obvious that if $[A,B]$ has full column rank, then $A$ and $B$ have full column rank.

But what if $[A,B]^T$ has full column rank. Then it doesn't imply that $A$ or $B$ have full column rank, right? (Also the converse direction doesn't work, right?)

Answer 1

Let $A=\begin{bmatrix}1 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$ and $B=\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}$. Then $[A,B]^\top$ has full column rank but $A$ does not.