## You are here

Homederivative of inverse matrix

## Primary tabs

# 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. ∎

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15-01*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag