Summary: The incidence matrices corresponding to a nil-algebra of finite index \(n\) can be used to determine the nilpotency. We find the smallest positive integer \(m\) such that the sum of the incidence matrices \(\sum_P \langle n,m \rangle^P\) is invertible. We give a different proof of the case that the nil-algebra of index \(2\) has nilpotency less than or equal to \(4\).