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Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible. Show that the inverse matrix $A^{-1}(t)$ is invertible and give a formula for the derivative of $A^{-1}(t)$ in terms of $A^{\prime}(t)$ and $A^{-1}(t)$. Thus one needs to investigate
$\lim _{h \rightarrow 0} \frac{A^{-1}(t+h)-A^{-1}(t)}{h}$
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