In the second half of the 1960s, Erdős proved that every singular matrix over a field can be expressed as a product of idempotent matrices. Since then, the characterization of integral domains satisfying the same property has become a widely investigated problem in ring theory. Notably, this problem is connected to other significant open questions, such as the characterization of integral domains whose general linear groups are generated by elementary matrices and those satisfying variations of the Euclidean algorithm. This seminar provides an informal overview of classical results regarding the idempotent factorization of matrices, as well as recent advancements in the field. Furthermore, it explores how the natural question "Can we study the (non-)uniqueness (in some sense) of idempotent matrix decompositions?" has led to a novel approach to factorization theory, significantly broadening the scope of the classical theory.