Determining the non-trivial matrix $A$, where $ AB = 0$.

Question

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 $A$?

Answer 1

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).