Rank inequality 
Let $ A, B \in M_n (\mathbb{C})$ such that $(A-B)^2 = A -B$. Then $\mathrm{rank}(A^2 - B^2) \geq \mathrm{rank}( AB -BA)$.

I tried to apply the basic inequalities without results. How to start? Thank you.

Hint: Let $P:=A-B$, then $A=P+B$, $P^2=P$ and
$$A^2-B^2= (P+B)^2-B^2=P^2+PB+BP=P+PB+BP\\
AB-BA=(P+B)B-B(P+B)=PB-BP$$
Now we can apply a basis transformation using an eigenbasis of $P$ so that the matrix of $P$ becomes diagonal with $k:={\rm rank}(P)$ ones and $n-k$ zeroes, so $PB$ gives the first $k$ columns of $B$ then $n-k$ zero columns, and similarly $BP$ gives the first $k$ rows of $B$ then $n-k$ zero rows.
