Show that the matrix $A+E$ is invertible. Let $A$ be an invertible matrix, and let $E$ be an upper triangular matrix with zeros on the diagonal. Assume that $AE=EA$. Show that the matrix $A+E$ is invertible. WLOG, we can assume $E$ is Jordan form. If $A$ is Jordan form, it's trivial. If $A$ is not Jordan form, how to use $AE=EA$ to transform $A$ to a Jordan form? Any suggestion? Thanks.

$E^n=0$ and since $A,E$ commute you have
$$A^{2n+1}=A^{2n+1}+E^{2n+1}=(A+E)(A^{2n}-A^{2n-1}E+...+E^{2n})$$
Since $A^{2n+1}$ is invertible, it follows that $A+E$ is invertible.
P.S. I only used in the proof that $E$ is nilpotent and commutes with $A$, so more generally it holds that in (any ring), if $A$ is invertible, $E$ is nil potent and $AE=EA$ then $A\pm E$ are invertible. 
