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Math Help - singular matrices 3

  1. #1
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    singular matrices 3

    if A is singular the A^3+A^2+A is singular
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  2. #2
    A Plied Mathematician
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    Note that A^{3}+A^{2}+A=A(A^{2}+A+I). Does that give you any ideas?
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  3. #3
    Behold, the power of SARDINES!
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    Yes, note that

    A(A^2+A+I) is a factorization and use the fact that

    \det(XY)=\det(X)\cdot \det(Y)

    too slow
    Last edited by TheEmptySet; November 16th 2010 at 12:37 PM. Reason: too slow
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  4. #4
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    how does determinat could tell if its singular?
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  5. #5
    A Plied Mathematician
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    \det(A)=0 if and only if A is singular.
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  6. #6
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    Quote Originally Posted by transgalactic View Post
    how does determinat could tell if its singular?
    What is the definition of "singular"?
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  7. #7
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    singular means that there could not be inverse
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  8. #8
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    You should have learned that det(AB)= det(A)det(B) (as TheEmptySet mentioned). If AB= I then det(A)det(B)= 1 so neither det(A) nor det(B) can be 0.

    However, Ackbeet's suggestion gives a more fundamental proof than that. If A^3+ A^2+ A had an inverse, then there would exist some matrix, B, such that (A^3+ A^2+ A)B= I . But that is the same as A[(A^2+ A+ I)B]= I which says that (A^2+ A+ I)B is inverse to A contradicting the fact that A does not have an inverse.
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