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Thread: Prove a cubed inverse matrix is the same as an inverse cubed matrix

  1. #1
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    Prove a cubed inverse matrix is the same as an inverse cubed matrix

    I have been asked to prove that the for a generic matrix A that, $\displaystyle (A^3)^{-1}$ is the same as $\displaystyle (A^{-1})^3$. I have no idea how to start this, any ideas would be much appreciated.

    Pretty much the wording is prove that A cubed, then inversed is the same as A inversed then cubed.
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  2. #2
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    $\displaystyle \left(A^3\right)^{-1}$ is, by definition, the matrix, B, such that $\displaystyle B(A^3)= (A^3)B= I$, the identity matrix. Show that this is true for $\displaystyle (A^{-1})^3A^3$ and $\displaystyle A^3(A^{-1})^3$.
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  3. #3
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    Done.

    Thank you very much, i know what to do. I'll do it just in case anyone else looks at this thread.
    First $\displaystyle (A^{-1})^3A^3$
    $\displaystyle (A^{-1})^3$ becomes $\displaystyle A^{-1}A^{-1}A^{-1}$
    $\displaystyle (A^3)$ becomes $\displaystyle AAA$
    and $\displaystyle AA^{-1} = I $ and $\displaystyle AI=A$ and $\displaystyle A^{-1}I=A^{-1}$
    Combining these facts makes:
    $\displaystyle AAAA^{-1}A^{-1}A^{-1}$ this becomes,
    $\displaystyle AAIA^{-1}A^{-1}$ which becomes,
    $\displaystyle AAA^{-1}A^{-1}$, after we do that a few times we get,
    $\displaystyle I$

    This also works for,
    $\displaystyle A^{-1}A^{-1}A^{-1}AAA$
    Please let me know if i did that right
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  4. #4
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    Yes, that's exactly what you needed to do.

    Now you use the fact that a matrix has at most one inverse to argue that $\displaystyle (A^{-1})^3= (A^3)^{-1}$.
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