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Homederivative of inverse matrix

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# 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. ∎

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