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Math Help - Prove the formula of inverse matrix

  1. #1
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    Prove the formula of inverse matrix

    In econometric, I have to use this formula to compute the inverse matrix
    (A\pm bc')^{-1}=A^{-1}\mp \left[\frac{1}{1\pm c'A^{-1}b}\right]A^{-1}bc'A^{-1}
    where A is a n\times n, symmetric, nonsingular matrix; b,c are n-column vectors, c' is transpose of c.
    I used it but I did know how to prove this formula. I need your help.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Have you tried


    (A\pm bc^t)\left(A^{-1}\mp \left(\frac{1}{1\pm c^tA^{-1}b}\right)A^{-1}bc^tA^{-1}\right)=\ldots=I\quad \textrm{?}
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  3. #3
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    I can prove it. But I can not explain it naturally. I do not know this formula is discovered from where.
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  4. #4
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    it is a special case of the Binomial Inverse Theorem, see here
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  5. #5
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    Quote Originally Posted by Deveno View Post
    it is a special case of the Binomial Inverse Theorem, see here
    Thank you so much.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mahefo View Post
    I can prove it. But I can not explain it naturally. I do not know this formula is discovered from where.

    Given A,B matrices, the natural way to prove the formula A^{-1}=B (that was your original question) is to prove AB=I . Another question is : given A , find A^{-1} .
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