> 
> Let $I$ be an identity matrix.
> 
> 
> If $I+A$ is a singular matrix (i.e. $I+A$ has at least one eigenvalue
>  being zero), then $\|A\|\geq 1$.
> 
> 
> 

How does a zero eigenvalue of $I+A$ imply $\|A\|\geq 1$?