Theorem 1   Suppose $A$ is a square matrix depending on a real parameter $t$ taking values in an open set . Further, suppose all component functions in $A$ are differentiable, and $A(t)$ is invertible for all $t$ . Then, in $I$ , we have $$ \frac{dA^{-1}}{dt}=-A^{-1} \frac{dA}{dt} A^{-1}, $$ where $\frac{d}{dt}$ is the derivative.

Proof. Suppose $a_{ij}(t)$ are the component functions for $A$ , and $a^{jk}(t)$ are component functions for $A^{-1}(t)$ . Then for each $t$ we have $$ \sum_{j=1}^n a_{ij}(t) a^{jk}(t)=\delta_i^k $$ where $n$ is the order of $A$ , and $\delta_i^k$ is the Kronecker delta symbol. Hence $$ \sum_{j=1}^n \frac{da_{ij}}{dt} a^{jk} + a_{ij}\frac{da^{jk}}{dt} = 0, $$ that is, $$ \frac{dA}{dt} A^{-1} = - A \frac{dA^{-1}}{dt} $$ from which the claim follows.