Showing that $A+\epsilon I$ is non-singular for $0<|\epsilon|

Question

How to approach this problem:

Let $A\in \mathcal{M}_n(\mathbb{R})$. Show that there exists a real number $r>0$ so that $A+\epsilon I$ is non-singular, when $\epsilon\in \mathbb{R}$, $0<|\epsilon|<r$

Thank you for any help.

Answer 1

Consider the function $p $ that sends $\epsilon $ to $det (A+\epsilon I) $.

$p $ is a polynomial of degree $n $. Hence it can be zero for at most $n $ values of $\epsilon $. Now choose $r>0 $ to have magnitude less than all of magnitudes of the non-zero roots of $p $