Suppose $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2 = P$, $Q^2=Q$ and $I-P-Q$ is invertible, where $I$ is the $n × n$ identity matrix. Show that $P$ and $Q$ have the same rank.
Since $I-P-Q$ is invertible, it has rank $n$. Also, $det(I-P-Q) \neq 0$. Can we get the result from these facts?
This seems to be homework, so I'll only give a hint.
Let $X = I - P - Q$. Now apply $P$ and $Q$ from the left and from the right to $X$ and see that happens.