# Inequality with matrix norms

Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then

$$\lVert (I-A)^{-1}\rVert \leq \frac{\lVert I\rVert-(\lVert I\rVert-1)\lVert A\rVert}{1-\lVert A\rVert}.$$

This is the second part of the problem. I finally figured out the first part, but I am having trouble starting this one. Does anyone know what inequality I should start with to get this? Or the procedure I should take? I think with a little help I should be able to figure this one out.

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What is $B$, all matrices of the same size? Could you also maybe tell us part 1 and how you solved it? –  bgins Apr 12 '12 at 16:04
Are you working with any matrix norm, or a particular one? –  bgins Apr 12 '12 at 16:13
First show that $\|(I-A)^{-1}\|\leq \frac{1}{1-\|A\|}$ –  Kuku Apr 12 '12 at 16:55
I'm working with just the standard norm, so it could be any norm. –  Sarah Martin Apr 12 '12 at 19:52