If A(t) is a continuously-differentiable $n \times n$ matrix function that is invertible at each $t$, show that:

$$ \frac{d}{dt}A^{-1}(t) = -A^{-1}(t)\,\dot{A}(t)\,A^{-1}(t) $$

First show that $t \mapsto A(t)^{-1}$ is also differentiable and then differentiate the identity $A(t) A(t)^{-1} = I_n$ (the identity matrix).

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