LetMbe an irreducible linear algebraic monoid defined overK(an algebraically closed field). LetMn(K) denote the set of alln×nmatrices overK. We writeNnfor the (irreducible closed) submonoid ofMn(K)) consisting of then×nmatrices of which each element can be expressed as a sum of a scalar matrix and a strictly upper triangular matrix.Mis nilpotent if its unit group is so. The following theorem is proved:Mis nilpotent iff ∃m,n1,n2,…,nm⩾ 1, such thatMis isomorphic to an irreducible closed submonoid of