National and Regional Contests Romania Contests Romania National Olympiad 2004 Romania National Olympiad 2 The inequality of the ranks Problem Statement Let n ∈ N n \in \mathbb N n ∈ N n ≥ 2 n \geq 2 n ≥ 2 A , B ∈ M n ( C ) A,B \in \mathcal M_n \left( \mathbb C \right) A , B ∈ M n ( C ) rank ( A B ) − rank ( B A ) = ⌊ n 2 ⌋ . \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . rank ( A B ) − rank ( B A ) = ⌊ 2 n ⌋ . X , Y ∈ M n ( C ) X,Y \in \mathcal M_n \left( \mathbb C \right) X , Y ∈ M n ( C ) rank ( X Y ) − rank ( Y X ) ≤ ⌊ n 2 ⌋ . \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . rank ( X Y ) − rank ( Y X ) ≤ ⌊ 2 n ⌋ .