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Present this matrix as a multiplication of elementary matrices question

  1. Feb 27, 2009 #1
    how to present this matrix as a multiplication of elementary matrices
    [tex]
    \bigl(\begin{smallmatrix}
    0 &6 &2 \\
    1& 1 &0 \\
    5&3 &1
    \end{smallmatrix}\bigr)
    [/tex]

    i cant understand in general what are they doing in this solution
    http://img144.imageshack.us/img144/9508/34036247.th.gif [Broken]

    ??
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 27, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I don't know what you mean by "I understand in general". Do you know how to "row-reduce" a matrix? Every "row-operation" corresponds to an elementary matrix: the matrix you get by performing that same row operation on the identity matrix.

    There are many different ways to row-reduce any matrix but I like to start on the left and clear one column at a time. For example, I might, as the first step in row-reduction, swap the first two rows, going from
    [tex]\begin{bmatrix} 0 & 6 & 2 \\ 1 & 1 & 0 \\ 5 & 3 & 1\end{bmatrix}[/tex]
    to
    [tex]\begin{bmatrix} 1 & 1 & 0 \\ 0 & 6 & 2 \\ 5 & 3 & 1\end{bmatrix}[/tex]

    Which corresponds to the elementary matrix we get by swapping the first two rows in the identity matrix:
    [tex]\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

    The next thing I might do is subtract 5 times the (new) first row from the third.
    [tex]\begin{bmatrix} 1 & 1 & 0 \\ 0 & 6 & 2 \\ 0 & -2 & 1\end{bmatrix}[/tex]
    and that corresponds to the elementary matrix
    [tex]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -5 & 0 & 1\end{bmatrix}[/tex]
    What we have done so far corresponds to the product of those two elementary matrices. Continue to row-reduce and write down the corresponding elementary matrix. Be careful about the order of multiplication.
     
  4. Feb 28, 2009 #3
    so when multiply all the elementary matrices
    i get the original one

    but they multiply only the first 4??

    and i cant understand whats E what F in the equations
    ??
     
  5. Feb 28, 2009 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    There are 6 elementary matrices. I don't see where they are multiplying "only the first 4".
    The "E" matrices are the elementary matrices corresponding to the row operations needed to row-reduce matrix A. The "F" matrices are their inverses (which are also elementary matrices). A is the product of the "F" matrices.
     
    Last edited: Mar 1, 2009
  6. Mar 1, 2009 #5
    i looked in a paper r and when there are operations like L1-5L2->L2
    then on the elemtary matrix part
    they say
    L1+5L2->L2
    i was told that its because they was to create elementary matrices which are invertable to itself X=X^-1
    E1*E2*E3*...*A=I
    so
    E1^-1*E2^-1*E3^-1..*E1*E2*E3*...*A=I*E1^-1*E2^-1*E3^-1..
    so
    A=E1^-1*E2^-1*E3^-1..
    but the inverses are the as the originals so we just need to multiply the
    elementary matrices
    correct??
     
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