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Prove that the matrices have the same rank.

  1. Oct 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that the three matrices have the same rank.


    [tex]

    \left[
    \begin{array}{c}
    A\\
    \end{array}
    \right]

    [/tex]

    [tex]

    \left[
    \begin{array}{c}
    A & A\\
    \end{array}
    \right]

    [/tex]

    [tex]

    \left[
    \begin{array}{cc}
    A & A\\
    A & A\\
    \end{array}
    \right]

    [/tex]

    2. Relevant equations



    3. The attempt at a solution

    If elimination is done on the second matrix it will become:
    [tex]

    \left[
    \begin{array}{c}
    A & 0\\
    \end{array}
    \right]

    [/tex]

    This means that the rank is still the same as A.

    Elimination on the third matrix gives:
    [tex]

    \left[
    \begin{array}{cc}
    A & A\\
    0 & 0\\
    \end{array}
    \right]

    [/tex]

    Since no new independent vectors are added, it also has rank A.

    I do understand that this is so, but could someone please help me explain this mathematically?

    Thanks.
     
  2. jcsd
  3. Oct 2, 2009 #2
    About the only thing different I would say is to suppose that when
    [tex]
    \left[
    \begin{array}{c}
    A\\
    \end{array}
    \right]
    [/tex]
    is reduced you obtain
    [tex]
    \left[
    \begin{array}{c}
    R\\
    \end{array}
    \right]
    [/tex]


    Then express your reduced forms of the other two matrices in terms of R instead of A. Refer to the number of nonzero rows in R, and you are done.
     
  4. Oct 4, 2009 #3
    Hi Billy Bob, thanks for the reply.
    Here's the way I think you would do it: (just showing one matrix)

    [tex]

    \left[
    \begin{array}{c}
    A & A\\
    \end{array}
    \right]

    [/tex]

    [tex]

    \left[
    \begin{array}{c}
    R & 0\\
    \end{array}
    \right]

    [/tex]

    # non zero rows = r for all matrices.
     
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