$P^2=P, Q^2=Q, (P+Q)^2=P+Q$, show $PQ=QP=0$.

Question

Let $P,Q$ be linear operator from $n$ dimensional vector space $V$ to $V$. If $P^2=P, Q^2=Q, (P+Q)^2=P+Q$, show $PQ=QP=0$.

It is easy to see $PQ+QP=0$. What to do next?

Answer 1

Multiply by $I-P$ on the right, to get $PQ(I-P)=0$, or $PQ=PQP$.

Multiply by $I-P$ on the left, to get $(I-P)QP=0$, or $QP=PQP$.