I am trying to solve the problem that is given below but I couldn't.
Suppose $A$ is $n\times n$ invertible matrix and $B$ is an $n\times n$ matrix such that $$\| A-B\| \leq \frac{1}{\| A^{-1}\|}.$$ Is $B$ invertible matrix.
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2What if $B=0$ and $A$ is the identity? – Arnaud Mortier Aug 03 '18 at 16:46
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If the inequality is strict $<$, then $AB=I-(I-AB)$ has inverse $\sum_{n=0}^{\infty}(I-AB)^n$, which converges because $|I-AB|<1$. – Aug 05 '18 at 22:20
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$B$ could be invertible: simply take $B=A$.
But it could also not be: take $B=0$ and $A=I_n$.
Arnaud Mortier
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