Let $A$ be an $n\times n$ singular matrix. Describe how to construct an $n\times n$ non zero matrix such that $AB = 0$.
I'm not sure how to approach this correctly. I think I am supposed to use the idea that $A$ is not invertible, which tells me that the columns of $A$ are linearly dependent and $A$ has less than $n$ pivots.
How can I use this information to construct $B$?
Guide:
Start by finding a non-zero $x$ such that $Ax=0$ (explain why you can find it).
Think of how to make good use of the $x$ that you find to construct $B$.