RO-NMO 2004 Problem 112

Let $n \in \mathbb N$ , $n \geq 2$ .

(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that $$\textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor .$$

(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have $$\textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor .$$

Ion Savu