It's a simple question that I expect will be closed as a duplicate, or answered with a link. That said, I've scoured the suggestions that came for my question title, as well as some relevant Wiki / math pages (even a few papers), and nothing really seems to treat this.
I feel that my understanding of eigenvalues and / or eigenvectors may be what is lacking, but from what I can tell, unless a matrix A = 0, then it will have at least one nonzero eigenvalue? Which would allow it to satisfy all of the conditions for a norm. Honestly a point in the right direction is all I'm looking for here.
Hint: Consider the matrices \begin{align} A= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \ \ \text{ and } \ \ B= \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \end{align}
What is the spectral radius of $A-B$?