I am working on the following exercise:
Consider a submultiplicative matrix norm $\| \cdot \|$. Let $A, B \in \mathbb{R}^{n \times n}$ such that $A$ is invertible. Assume that
$$\|B-A\| < \frac{1}{\|A^{-1}\|}.$$
Show that $B$ is invertible.
I do not know what I could do to prove that. Could you give me a hint? Does it have something to do with eigenvalues or the spectral theorem?
Hint: Note that $$ \|A^{-1}B - I\| = \|A^{-1}(B-A)\| \leq \|A\|^{-1}\cdot \|B-A\| < 1. $$ From there, one approach is to consider the eigenvalues of $A^{-1}B$.