Two idempotent matrix

Question

Let $A,G$ be two $n\times n$ matrix satisfying: $$A^2=A, GAG=G, im(G)\subset im(A).$$ Prove that $G^2=G$.

I do not know how to prove it.

Answer 1

$im(G) \subset im(A)$,so for any vector $u$,there is a vector $v$ that

$$Gu=Av$$.

So we get:

$$Gu=GAGu=GAAv=GA^2v=GAv=GGu=G^2u$$

.

QED.