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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