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Math Help - Proof

  1. #1
    patrick
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    Proof

    Hello,
    For a nonsingular matrix A and a nonnegative integer p, show what (A^p)^-1 = (A^-1)^P
    How to proof it.
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  2. #2
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    Quote Originally Posted by patrick View Post
    Hello,
    For a nonsingular matrix A and a nonnegative integer p, show what (A^p)^-1 = (A^-1)^P
    How to proof it.
    You should also say A is a square.

    (A^p)^{-1} = (AA...A)^{-1}
    In general (even for a group) for matrices (invertible):
    (AB)^{-1} = B^{-1}A^{-1}.
    Use this rule repeatedly here to get,
    A^{-1}A^{-1}...A^{-1} = \left(A^{-1}\right)^p.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by patrick View Post
    Hello,
    For a nonsingular matrix A and a nonnegative integer p, show what (A^p)^-1 = (A^-1)^P
    How to proof it.
    (A^p)^{-1} is the multiplicative inverse of the matrix (A^p). So we know that
    (A^p)^{-1} \cdot (A^p) = I

    Furthermore, this inverse is unique.

    So take (A^{-1})^p and multiply it by (A^p). If the result is I, then the proposition is true.
    (A^{-1})^p \cdot (A^p) = (A^{-1} \cdot A)^p = (I)^p = I

    Therefore etc.

    -Dan
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