From an assignment:
Let $A = \left[ \begin{matrix} 3 & -6 \\ -2 & 4 \end{matrix}\right] $ Construct a $2 * 2$ matrix $B$ such that $AB$ is the zero matrix. Use two different nonzero columns for $B$.
The value of $AB$ would be:
$$ AB = \left[ \begin{matrix} 3b_{11} -6b_{12} & 3b_{21} -6b_{22} \\ -2b_{11} + 4b_{12} & -2b_{21} + 4b_{22} \end{matrix}\right] $$
I was thinking of using substitution, but the following equations just result in the variables equalling $0$:
$$\begin{align*} 3b_{11} -6b_{12} &= 0\\ -2b_{11} + 4b_{12} &= 0 \end{align*}$$
Any hints on how I can solve this?