Homogenous Systems Question $A, B \in \mathbb{C}^{n\times n}$
If $I - B$ is singular, then $\exists x$ from $\mathbb{C}^n$  so that $(I - B)x = 0$. 
How do I show that $|| B || \geq 1$ and if $|| A || \leq 1$ then $I-A$ isn't singular?
It's not really clear for me either, but I think the norm I'm supposed to use is $ || A || = \max \{ | a_{ij} | \}$ 

The correct statement should be:

if $\|A\|\color{red}{<}1$, then $I-A$ is nonsingular.

The usual proof is to show that the power series $I+A+A^2+\ldots$ converges (we need $\|A\|<1$ here) and it is the inverse of $I-A$. Which norm you are using is irrelevant, as long as the norm is a submultiplicative matrix norm.
Your statement for $B$ is just the contrapositive of the statement in the grey box.
