Solution
= Solution
For a differentiable family of invertible <linear operators>, differentiate the identity without changing the factor order:
$$
\frac{dA^{-1}}{d\lambda}A+A^{-1}\frac{dA}{d\lambda}=0.
$$
Multiplication on the right by $A^{-1}$ gives the <derivative of matrix inversion>,
$$
\boxed{\frac{dA^{-1}}{d\lambda}=-A^{-1}\frac{dA}{d\lambda}A^{-1}.}
$$
Commuting $A^{-1}$ past the <derivative> would generally be invalid.