The sum of a normal and nilpotent matrix.

Question

Assume that $n \times n$ matrices $A$ and $B$ are such that $A$ is normal, $B$ is nilpotent, and $A + B = I$. Prove that $A=I$.

Answer 1

$B$ is normal, since $$BB^\dagger=(I-A)(I-A)^\dagger=I-A-A^\dagger-AA^\dagger$$ $$B^\dagger B=(I-A)^\dagger(I-A)=I-A^\dagger-A-A^\dagger A$$ and $A$ is normal, so by $A$ is normal and nilpotent, show $A=0$, $B=0$ and $A=I$.