1. For , let be the subspace of spanned by the first k vectors in the standard basis. The condition for A to be upper triangular is that for each k. If A is invertible then its kernel consists only of the zero vector. So has the same dimension as and therefore for each k. Thus , which says that is upper triangular.

2. If some power of A is 0 then the "binomial series" becomes a finite series series which gives you a formula for .