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Math Help - matrice 2

  1. #1
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    matrice 2

    (1)IF A and B are square matrices such that AB=\omega A and BA=\phi B where \phi and  \omega are non - zero scalars , prove that A^2=\phi A and B^2=\omega B .

    (2) If A is a 2x2 matrix such that A^2=\omega A where \omega is a non-zero scalar , prove that A is either non-singular or A=\phi I where I is an identity .


    Thanks for helping me out .
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  2. #2
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    Quote Originally Posted by thereddevils View Post
    (1)IF A and B are square matrices such that AB=\omega A and BA=\phi B where \phi and  \omega are non - zero scalars , prove that A^2=\phi A and B^2=\omega B .
    If AB=\omega A then A = \omega^{-1}AB, so A^2 = (\omega^{-1}AB)A =\omega^{-1}A(BA) =\omega^{-1}A(\phi B) = \ldots.

    Quote Originally Posted by thereddevils View Post
    (2) If A is a 2x2 matrix such that A^2=\omega A where \omega is a non-zero scalar , prove that A is either non-singular or A=\phi I where I is an identity.
    I think you mean "singular", not "non-singular". If A is non-singular then you can multiply both sides of the equation A^2=\omega A by A^{-1} and conclude that A = \omega I.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    If AB=\omega A then A = \omega^{-1}AB, so A^2 = (\omega^{-1}AB)A =\omega^{-1}A(BA) =\omega^{-1}A(\phi B) = \ldots.


    I think you mean "singular", not "non-singular". If A is non-singular then you can multiply both sides of the equation A^2=\omega A by A^{-1} and conclude that A = \omega I.
    Thanks Opalg, but how can i continue from here \omega^{-1}A(\phi B) = \ldots , how to get rid of some variables here to get the required result .
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  4. #4
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    Quote Originally Posted by thereddevils View Post
    Thanks Opalg, but how can i continue from here \omega^{-1}A(\phi B) = \ldots , how to get rid of some variables here to get the required result .
    I thought you'd be able to manage the rest of it.

    \omega^{-1}A(\phi B) = \phi(\omega^{-1}AB) = \phi A.
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  5. #5
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    Quote Originally Posted by Opalg View Post
    I thought you'd be able to manage the rest of it.

    \omega^{-1}A(\phi B) = \phi(\omega^{-1}AB) = \phi A.

    gosh , sorry to disappoint u ..
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