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Thread: Matrix Inverse Proof

  1. #1
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    Matrix Inverse Proof

    Show that $\displaystyle (A ^{-1})^T = (A ^T)^{-1}$

    Assuming A is invertible.
    Last edited by Jameson; Sep 9th 2009 at 01:55 PM.
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  2. #2
    MHF Contributor Matt Westwood's Avatar
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    If you can take the assumption that the determinant of a matrix and its transpose are equal, you can do it by induction and the definition of the inverse as it is defined in terms of cofactors.

    Here's Wikipedia:

    Invertible matrix - Wikipedia, the free encyclopedia

    ... it might help.
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  3. #3
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    $\displaystyle A A^{-1} = I $

    $\displaystyle (A A^{-1})^T = (I)^T$

    $\displaystyle (A^{-1})^T A^T = I$

    By definition of inverse matrix, $\displaystyle (A^{-1})^T$ is the inverse of $\displaystyle A^T$, hence:

    $\displaystyle (A^{-1})^T = (A^T)^{-1}$
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