We know that if $A,B$ idempotent then we have (see edit) $$(A+B)^2=A+B\implies AB=0,$$
and I'm wondering whether the converse, i.e. that if $A,B$ idempotent and $AB=0$ then $A+B$ idempotent is true. Expanding we get
$$(A+B)^2=A^2+AB+BA+B^2=A+B+BA$$
so we just need to show that $BA=0$. It certainly isn't true in general that $AB=0\implies BA=0$, but I couldn't think of an example where $A$ and $B$ are idempotent. For orthogonal projections at least I think that the statement is true (by thinking geometrically), but I need to consider general projections.
My intuition is that this statement probably is true. Is this correct, and if so how can I show this?
Edit:
$(A+B)^2=A+B\implies AB=-BA$, but then $BAB=-BA$ and $AB=-BAB$, so $AB=BA$ and therefore $AB=BA=0$.
Best Answer
What if $A =\left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)$ and $B =\left(\begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix}\right)$?
$AB = 0$, but $BA = \left(\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}\right)$.
And $A+B = \left(\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}\right)$ is not idempotent since $(A+B)^2 = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right)$.