Difference between $\|A^n\|$ and $\|A\|^n$?

Question

In matrix norms, what is the difference between $\|A^n\|$ and $\|A\|^n$?

Answer 1

For example, let $A=({}^0_0\,{}^2_0)$ and suppose we're talking the operator norm.

To compute $\|A^2\|$ we first take the square of $A$, giving $({}^0_0\,{}^0_0)$, and take the norm of that, which is $0$.

To compute $\|A\|^2$, we first take the norm of $A$, which is $2$, and square that, giving $4$.