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Math Help - Proving Matrix

  1. #1
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    Proving Matrix

    Sorry for this second question!

    How do I prove that A^2 = A in Matrices given that AB=A and BA=A.
    I tried to expand the LHS and then intend to get at AB so as to end up with RHS=A in the end but I'm stuckm
    A^2 = A
    AA = A
    A(AB) = A
    ...

    If I keep expandin, I'll keep getting more Bs and As..
    Please help me.

    Thank you!
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  2. #2
    Junior Member Barioth's Avatar
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    Re: Proving Matrix

    Quote Originally Posted by Tutu View Post
    Sorry for this second question!
    Don't be sorry it's a fun one!

    I had start out saying that for A=0 it is pretty trivial that it work.

    for A diff of 0 now, B can not be 0 since AB=A.

    I'm pretty sure you can't go using only the matrice themself. There is no way you can prove that B or A as an inverse.
    I think that you'll have to get your hand dirty and use the value in the matrice. Since we dont know the size you'll also have to work with a (NxN) matrices. You should first try with a 2x2 matrice. (I'm pretty sure it could work there if such a proof exist.)

    Good luck!
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  3. #3
    Senior Member vincisonfire's Avatar
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    Re: Proving Matrix

    You are given AB=A and BA=A (for any matrix B I assume). Substitute B by A.
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  4. #4
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    Re: Proving Matrix

    Thank you both but I'm still stuck, can someone show me?

    A^2
    =AA
    =A(AB)
    A(A(BA))
    = A^2(BA)
    ?
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  5. #5
    Junior Member Barioth's Avatar
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    Re: Proving Matrix

    Quote Originally Posted by vincisonfire View Post
    (for any matrix B I assume).
    Tutu is it for any matrix B? if so, A as to be the nul matrix. What is the full question?
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  6. #6
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    Re: Proving Matrix

    Quote Originally Posted by Tutu View Post
    Sorry for this second question!

    How do I prove that A^2 = A in Matrices given that AB=A and BA=A.
    I tried to expand the LHS and then intend to get at AB so as to end up with RHS=A in the end but I'm stuckm
    A^2 = A
    AA = A
    A(AB) = A
    ...

    If I keep expandin, I'll keep getting more Bs and As..
    Please help me.

    Thank you!
    Hi Tutu,

    It is not true in general that if there are matrices A and B with AB = A and BA = A then A^2 = A.

    For example, let

    A = \begin{pmatrix} 1 &0\\ 0 &2 \end{pmatrix} \quad B = \begin{pmatrix} 1 &0\\ 0 &1 \end{pmatrix}
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  7. #7
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    Re: Proving Matrix

    Hi thank you all! Sorry for not including the whole question, didn't realize it was that important, my fault and apologies.

    It is known that AB=A and BA=B where the matrices A and B are not necessarily invertible. Prove that A^2 = A (Note: From AB = A, you cannot deduce that B=I. Why?)

    Adapted from IB Math HL Haese and Harris Publications
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