Notions of topological spaces and matrices

Question

How can I show that:

1. For all $r>0$, exists $A$ nonsingular matrix, such that $B_r(A)\subseteq GL_n(\mathbb{C})$
2. For all $A\in GL_n(\mathbb{C})$ and for all $r>0$, there exists $\alpha>0$ such that $B_r(\alpha A)\subseteq GL_n(\mathbb{C})$.

We define $B_r(A)=\{X\in M_n : ||A-X||_F<r\}$ and $||A||_F=\textrm{Tr}(A^{\star}A)^{{1\over 2}}$

Answer 1

Hint: Consider the Neumann-series: $$ (\def\Id{\mathop{\rm Id}}\Id - A)^{-1} = \sum_{n=0}^\infty A^n $$ which converges for $\def\norm#1{\left\|#1\right\|_F}\norm A < 1$. So if $\norm{\Id - A} < 1$, $A$ is invertible. If now $A$ is any invertible matrix, we have for $B \in M_n(\mathbb C)$: $$ B - A = B(\Id - B^{-1}A) $$ that is if $\norm{B-A} < \norm{B^{-1}}^{-1}$, then $B^{-1}A$ and hence $A$ is invertible.