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Math Help - AB=A+B;Prove A and B commute

  1. #1
    Senior Member pankaj's Avatar
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    AB=A+B;Prove A and B commute

    If A and B are two square matrices of the same order such that AB=A+B then prove that A and B commute i.e. AB=BA
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  2. #2
    Super Member Anonymous1's Avatar
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    Umm working on it.

    AB = A+B

    A(BB^{-1}) = (A + B)B^{-1}

    A = (AB^{-1} + I)

    BA = BAB^{-1} + B


    ...
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  3. #3
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    Quote Originally Posted by pankaj View Post
    If A and B are two square matrices of the same order such that AB=A+B then prove that A and B commute i.e. AB=BA
    If AB=A+B then (A-I)(B-I)=I (I is the identity matrix). So B-I is the inverse of A-I and therefore commutes with it. Thus (B-I)(A-I) = I, from which BA = A+B = AB.
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  4. #4
    Senior Member pankaj's Avatar
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    So if CD=I does it always mean that C and D are inverse of each other.What if any one of C or D is singular matrix
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    MHF Contributor
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    Quote Originally Posted by pankaj View Post
    So if CD=I does it always mean that C and D are inverse of each other.What if any one of C or D is singular matrix
    If CD = I then C and D cannot be singular. One way to see this is to notice that the determinant is multiplicative: \det(C)\det(D) = \det(I) = 1, so that \det(C) and \det(D) cannot be 0.
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