RO-NMO 2006 Problem 112

We define a pseudo-inverse $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations $$A = ABA \text{ and } B=BAB.$$ a) Prove that any square matrix has at least a pseudo-inverse.

b) For which matrix $A$ is the pseudo-inverse unique?

Marius Cavachi