Determining the non-trivial matrix $A$, where $ AB = 0$. Example:
$A$ is a $3\times 4$ matrix.
$B$ is the following $4\times 2$ matrix:
\begin{pmatrix}
1 & 0 \\
0 & 2 \\
1 & 1 \\
1 & 2 \\
\end{pmatrix}
$A$ times $B$ equals a zero matrix.
How do I find the non-trivial solution to $A$?

A simple way to do this is to note you have 2 vectors for the columns of $B$. Thus, if you pick 3 vectors orthogonal to the columns of $B$ (say, by finding 2 non-zero vectors orthogonal to the columns of $B$ via gram schmidt and the zero vector), then the matrix consisting of those 3 vectors as its rows satisfies this. 
Note that mixing the rows (i.e. forming linear combinations of the prescribed rows) will give you another valid $A$. 
(You should be thinking in terms of left null spaces). 
