# Condition number inequality for non-signular matrices

Let $$A \in \mathbb{R}$$ be non-singular. Assume that for some induced matrix norm $$\frac{||E||}{||A||} \leq \frac{1}{\kappa(A)}$$ Prove that $$A+E$$ is non-singular, where $$A + E$$ is the perturbation of A and $$\kappa(A)$$ is the condition number of A.

How can I go about proving the above statement? I'm new to numerical linear algebra and would appreciate any help on this problem.

I assume you mean $$A\in \mathbb{R}^{n\times n}$$ and that the inequality is strict (otherwise the statement is wrong). We want to show that $$A+E$$ is non-singular, i.e., that $$A+E$$ is invertible. Since $$A$$ is assumed to be invertible, we can write
$$A+E=A(I+A^{-1}E)$$, so it remains to show that $$(I+A^{-1}E)$$ is invertible. But for any $$B\in \mathbb{R}^{n\times n}$$, we know by the Neumann series that $$(I+B)$$ is invertible as long as $$\|B\|<1$$. But by assumption, we have
$$\|A^{-1}E\|\leq \|A^{-1}\| \|E\|=\kappa(A) \frac{\|E\|}{\|A\|}<\kappa(A)\frac{1}{\kappa(A)}=1$$.
Thus $$(I+A^{-1}E)$$ is invertible, which yields the desired claim.