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Math Help - Row-reduced echelon matrix

  1. #1
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    Row-reduced echelon matrix

    Suppose R and S are 2x3 row -reduced echelon matrices and that the systems RX=0 and SX=0 have exactly the same solutions. Prove that R=S.
    Can anyone give me any hint to start the proof?Thanks.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Row-reduced echelon matrix

    Try by contradiction: if R\neq S then ...
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    Re: Row-reduced echelon matrix

    Can I claim that suppose one of the entries in both R and S are not the same, then using the same solution, claim that there is a contradiction and actually the entry(that I had supposed not the same) is the same?

    That is suppose both R and S have the same second row but different first row with the 1,1 and 1,2 entries the same but 1,3 entry differs. Let the 1,3 entries of R and S be r and s respectively. Then if we have a solution x_1,x_2 and x_3, we will have rx_3=sx_3. But I am stucked here as if  x_3 is zero, then my argument does not create contradiction.
    Last edited by problem; August 30th 2011 at 07:42 AM.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Row-reduced echelon matrix

    If \textrm{rank}(R)\neq \textrm{rank}(S) then, \textrm{Nul}(R)\neq \textrm{Nul}(S) because they have different dimensions. Analyze the cases \textrm{rank}(R)= \textrm{rank}(S) . For example R=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{0} \end{bmatrix} and S=\begin{bmatrix}{0}&{1}&{0}\\{0}&{0}&{0} \end{bmatrix}
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    Junior Member bondesan's Avatar
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    Re: Row-reduced echelon matrix

    I think that if x\neq 0 then RX-SX=0 => X=0 or R-S=0, which means that R=S because of our assumption that x\neq 0.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Row-reduced echelon matrix

    Quote Originally Posted by bondesan View Post
    I think that if x\neq 0 then RX-SX=0 => X=0 or R-S=0, which means that R=S because of our assumption that x\neq 0.
    That is not true, choose for example R=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{0} \end{bmatrix} , S=\begin{bmatrix}{0}&{1}&{0}\\{0}&{0}&{0} \end{bmatrix} and x=\begin{bmatrix}{0}\\{0}\\{1} \end{bmatrix}
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    Re: Row-reduced echelon matrix

    Hi FernandoRevilla,

    The R and S that you had given are in row-reduced form and the X that is given by you is also a solution for RX=0 and SX=0. Does this mean that what I want to prove(the initial question) is not correct?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Row-reduced echelon matrix

    Quote Originally Posted by problem View Post
    The R and S that you had given are in row-reduced form and the X that is given by you is also a solution for RX=0 and SX=0. Does this mean that what I want to prove(the initial question) is not correct?
    No, because (for example) \begin{bmatrix}{0}\\{1}\\{0} \end{bmatrix} is a solution of Rx=0 but not of Sx=0 i.e., the systems have not the same solutions.
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