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Show that if $A+delta A$ is singular, $|delta A| = |A^{-1}|^{-1}$

Mathematics Asked by Flerovium on January 12, 2021

I’m trying to show the following.

Let $A$ be an invertible $ntimes n$-matrix and $delta A$ be the smallest possible matrix, measured in a subordinate (natural) matrix norm $|cdot|$ such that $A+delta A$ is singular. Then

$$
|delta A| = |A^{-1}|^{-1}
$$

I’m a bit lost as to where one should start on such a problem. Any suggestions?

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