Let $$ X = \begin{bmatrix} A & 0 \\ C & D \end{bmatrix}$$ where the number of rows of $A$, $C$ and $D$ are larger than those of columns.
If both $\begin{bmatrix} A \\ C \end{bmatrix}$ and $\begin{bmatrix} C & D \end{bmatrix}$ are of full column rank, is $X$ also of full column rank?
Or, can we say that $X$ is of full column rank if $\begin{bmatrix} C & D \end{bmatrix}$ are of full column rank?