Is it possible $\lVert a\rVert =\lVert 1+a\rVert $ in a C$^*$-algebra?

Question

If $A$ is a unital C$^*$-algebra and $a\in A$, Is it possible $ \lVert a \rVert =\lVert 1+a \rVert $ for an $a\geq 0$ ?

I think it's trivial that it's not possible but I can't prove it for even $ A=Mat_{n\times n}(\mathbb{C})! $

Answer 1

It is certainly not possible for $a\geq 0$ since $\lVert x\rVert=\sup \sigma(x)$ for $x\geq 0$ and $\sigma(1+a)=1+\sigma(a)$.