Let *B*, *A*, and *e* be square matrices with *e* small, and define

(1) |

(2) |

(3) |

Note that if we instead let
, and look for an inverse of the form
, we obtain

(4) |

In order to eliminate the term, we require . However, then , so so there can be no inverse of this form.

The exact inverse of
can be found as follows.

(5) |

(6) |

(7) |

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1999-05-26