Specific Case For Jordan Block

Question

Let $F$ be a field, and take $\lambda\in F$ with $n\geq 1$. Let $A = (a_{ij})\in M_n(F)$ be an upper triangular matrix, where $a_{ii}=\lambda$ for all $i$. Show that the Jordan Canonical Form of $A$ has a single block if and only if $a_{12}a_{23}....a_{(n-1)(n)}$ is nonzero. I'm unsure of how to use the fact that $A$ is upper triangular; would Schur's lemma be helpful here?

Answer 1

Hint: If $J$ is the Jordan form of $A$, then $A - \lambda I$ and $J - \lambda I$ will have the same rank.