Let us consider \begin{equation} D= \begin{bmatrix} A & 0 \\ B & C \end{bmatrix} \end{equation} where $Ax \neq 0$ with $x \neq 0$ and the number of rows of $A$ can be larger than that of columns.
In this case, does the following equality hold? \begin{equation} rank(D) = rank \begin{bmatrix} A \\ B \end{bmatrix} + rank[C] \end{equation}
My conjecture is that since $Ax \neq 0$ with $x \neq 0$, the column space spanned by $\begin{bmatrix} A \\ B \end{bmatrix} $ does not lie on that of $\begin{bmatrix} 0 \\ C \end{bmatrix}$ and hence we obtain the above result. Is this correct?