TITLE: Finding matrix $B$ is not zero matrix where $AB= 0$
QUESTION [5 upvotes]: $A$ is defined as an $m\times m$ matrix which is not invertible. How can i show that there is an $m\times m$ matrix $B$ where $AB = 0$ but $B$ is not equal to $0$? 
For the solution of this question I think giving an example is not enough because it is too easy to solve this by giving an example, so how can I show that $B$ is not the $0$ matrix?

REPLY [3 votes]: Since $A$ is not invertible, by the Invertible Matrix Theorem there is a nontrivial solution
$x_0$ of the homogeneous equation $Ax_0 = 0$. Take $x_0$ as the first column of $B$
and take the remaining columns of $B$ to be zero. That gives $AB=0$ but $B$ is not the zero matrix.