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Thread: Intro Linear Algebra proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Intro Linear Algebra proof

    If A is a nxn matrix that is idempotent and invertible, then $\displaystyle A=I_{n}$.

    Let A be any nxn matrix such that A is idempotent and invertible. Then, by definition A satisfies the following properties,

    $\displaystyle A^{2}=A$ and A can be written as the product of elementary matrices.

    So, since $\displaystyle A^{2}=A$ we can say $\displaystyle A^{2}=E_{k}^{-1}...E_{1}^{-1}$

    ...I don't really know how to finish this...

    Should I use an equivalent condition for invertible matrices?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    $\displaystyle A^2=A$

    $\displaystyle A^{-1}A^2=A^{-1}A$ (A is invertible)

    $\displaystyle A=I$
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Wow...I feel dumb. Thanks alot. I figured I had to use the elementary matrices since thats all the section was about ha.
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