Question: Let A be a n by n complex matrix and ||.||p be the matrix p-norm for p in [1, infinity]. Suppose A is invertible. Show

Let A be a n by n complex matrix and ||.||p be the matrix p-norm for p in [1, infinity].

Suppose A is invertible. Show that for all n by n complex matrix X with

|| X-A ||p < 1 / || A^-1 ||p must be invertible.

(I'm struggling with this question. I was trying to assume A is not invertible and prove it by the contradiction, but didn't end up with a solution. Can you show me how to approach this question?)

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