TITLE: $AB \neq 0$ but $BA=0$
QUESTION [10 upvotes]: Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$?
Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that...
$[A,B]=AB$
where $[ \, , \, ]$ is the commutator $[A,B]=AB-BA.$
If such matrices do not exist, what does that imply about the algebra that the elements are in?

REPLY [4 votes]: As the other answer show: there are uncountably many pairs of matrices $(A,B)$ such that $AB = 0$ while $BA \neq 0$. If you think of square matrices as linear transformations then it is obvious why this should be so: in $AB$, we can think of the product as recording the image of each of the columns of $B$ under the linear transformation $A$, likewise with $BA$.
A nice question to ask is the following: Given a fixed matrix $A \in \text{Mat}_n\mathbb{R},$ what is the following:
$$\widetilde{A} := \{ X \in \text{Mat}_n\mathbb{R} : AX = 0 \ \wedge \ XA \neq 0\} \, ?$$
Clearly, if $X \in \widetilde{A}$ then $\lambda X \in \widetilde{A}$ for all $\lambda \neq 0.$ Interestingly, this space is not a vector space because the zero matrix $0 \notin \widetilde{A}$ and $X,Y \in \widetilde{A}$ does not imply that $X+Y \in \widetilde{A}.$