Is it possible to have a matrix norm on $M_n$ where $n>1$ such that $\|AB\| = \|A\|\|B\|$ for all $A,B\in M_n$?
I am very inclined to say no, there does not exist a matrix norm where equality is always possible. We know that by definition of a matrix norm, $\|X\| = 0 \iff X = O$. Consider some matrices $$A=\begin{bmatrix}1 & 0 & 0 & \cdots & 0\\0 & 0 & 0 & \cdots &0\\ \vdots & \vdots & \vdots & \ddots &\vdots\\0 & 0 & 0 & \cdots &0\end{bmatrix},\;\;\; B = \begin{bmatrix}0 & 0 & 0 & \cdots & 0\\0 & 0 & 0 & \cdots &0\\ \vdots & \vdots & \vdots & \ddots &\vdots\\0 & 0 & 0 & \cdots &1\end{bmatrix} \in M_n.$$
Obviously, $A\ne O$ and $B\ne O$, but $AB = O$. Hence $\|AB\| = \|O\| = 0$, but since $A,B \ne O$, then $\|A\|,\|B\| >0$ which implies $\|A\|\|B\| > 0$ which is not zero.
Is this valid reasoning to believe no such norm exists over $M_n$?
My professor was somewhat vague saying that "usually submultiplicativity equality doesn't happen." Why usually? I unfortunately did not get a very satisfactory answer when I asked about the "usually" part. That might imply there does exist, but logically one doesn't seem to exist. Am I missing something here?