Factorizations of integer matrices as products of idempotents and nilpotents (Q1123235)

Let A be a singular \(n\times n\) matrix with integer entries. Then for \(n\geq 3\), A is the product of \(36n+217\) idempotent and the product of \(72n+434\) nilpotent matrices with integer entries. For \(2\times 2\) matrices there is no such upper bound. It is shown that to any given positive integer N there exist \(2\times 2\) matrices A, B with integer entries such that A is the product of N but no fewer idempotent and B is the product of N but no fewer nilpotent matrices with integer entries.