Is there a matrix $N$ such that $M\cdot N$ is of the form..

Question

suppose I have a matrix $M\in M_{n}(\mathbb R)$ in block form, $$M=\left[\begin{array}{cccccc}A &|&B\\ \hline C&|&D\end{array}\right], $$ where $A\in M_{k}(\mathbb R)$. Is there a matrix $N$ such that, $$M\cdot N=\left[\begin{array}{cccccc}A &|&B\\ \hline 0&|&0\end{array}\right].$$ It is allowed transposing $M$, $N$ or using this matrices more then once in the product. Thanks, any help will be valuable..

Answer 1

No, generally speaking. E.g., $2\times 2$-matrix $M=\left[\begin{array}{cc}1 &1\\ 1&1\end{array}\right]$.