# If A(t) is a continuously-differentiable n×n matrix function that is invertible at each t, show that

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).