$2$ square matrices with equal rank

Question

Suppose $P$ and $Q$ are $n\times n$ matrices of real numbers such that

• $P^2=P$
• $Q^2=Q$
• $I−P−Q$ is invertible, where $I$ is a $n\times n$ identity matrix.

Show that $P$ and $Q$ have the same rank.

Answer 1

HINT: $$rank(P)=rank(P(I-P-Q)),\\rank(Q)=rank((I-P-Q)Q)$$