Does the spectral radius of a matrix define a norm?

Question

Does the spectral radius of a matrix define a norm? Does it satisfy the properties of norm, i.e.,

1. $\| x \| \ge 0$
2. $\| x \| = 0 \implies x=0$
3. $\| kx \| = |k| \cdot \|x \|$
4. $\| x+y \| \le \|x\| + \|y\|$

Answer 1

Choose matrices x and y such that the fourth property (triangle inequality) is not satisfied. There are plenty of matrices which will show that

$$\rho(A+B) > \rho(A)+\rho(B)$$

Answer 2

if you choose any nilpotent matrix M so you have existence of $k\in\mathbb{N}$ such that $M^k=0$ so in $M^n=0$ for all $n\geq k$ and : $$ \|M^n\|^{1/n}=0 $$ So $\rho(M)=\lim_n \|M^n\|^{1/n}=0$. we can conclude that the spectral radius of any nilpotent matrix is equal to $0$. and so the spectral radius can't be a norme.