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Math Help - 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, (A^3)^{-1} is the same as (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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    \left(A^3\right)^{-1} is, by definition, the matrix, B, such that B(A^3)= (A^3)B= I, the identity matrix. Show that this is true for (A^{-1})^3A^3 and 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 (A^{-1})^3A^3
    (A^{-1})^3 becomes A^{-1}A^{-1}A^{-1}
    (A^3) becomes AAA
    and AA^{-1} = I and AI=A and A^{-1}I=A^{-1}
    Combining these facts makes:
    AAAA^{-1}A^{-1}A^{-1} this becomes,
    AAIA^{-1}A^{-1} which becomes,
    AAA^{-1}A^{-1}, after we do that a few times we get,
    I

    This also works for,
    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 (A^{-1})^3= (A^3)^{-1}.
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