# derivative of inverse matrix

###### Theorem 1.

Suppose $A$ is a square matrix depending on a real parameter $t$ taking values in an open set $I\subseteq\mathbbmss{R}$. 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. ∎

Title derivative of inverse matrix DerivativeOfInverseMatrix 2013-03-22 14:43:52 2013-03-22 14:43:52 matte (1858) matte (1858) 7 matte (1858) Theorem msc 15-01