# Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove invertibility of matrices (or at least, tridiagonal ones).

Could someone point me towards anything less complicated than in-depth mathematical proofs to help me to understand what banach's lemma actually tells about matrices (and how that proves invertibility?)

edit: realizing that the definition as "banachs lemma" was somewhat confusing, here's the lemma that I'm referring to:

if in some induced matrix norm $||A|| < 1$ (where A is an $n * n$ matrix)

then $I + A$ is nonsingular and $$||(I+A)^{-1}|| \le \frac{1}{1 - ||A||}$$

edit2: it seems to boil down to using the $||A||_2$ basis (eigenvalues) of tridiagonal matrices to induce a norm that is less than 1, which can then be used to prove invertibility, but still trying to work this out!

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