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Given an invertible matrix $M$ which depends on a real parameter $\alpha$. Is the limit $\lim_{\alpha - \alpha_0} \left( M^{-1} \right) $ same as $ \left( \lim_{\alpha - \alpha_0} M\right)^{-1}$, i.e. does the following hold

\begin{equation} \lim_{\alpha - \alpha_0} \left( M^{-1} \right) \overset{\mathrm{?}}{=} \ \left( \lim_{\alpha - \alpha_0} M\right)^{-1} \end{equation} Or is there any relation between these quantities (say one is less or equal to the other)?

Rob
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    The equality holds, provided $\lim _{a\to a_0} M(a)$ is invertible. So excluding stuff like $aE,\ a\to 0$. – AlvinL Aug 04 '23 at 08:31

1 Answers1

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The map $M\mapsto M^{-1}$ is continuous on $\mathrm{GL}_n(\mathbb{R})$, see here. Hence the answer is affirmative.

Paolo Leonetti
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