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Matrix operations questions

  1. Jun 17, 2014 #1
    Can I get this matrix
    \begin{smallmatrix}
    1&1&0\\ 0&0&1\\ 0&0&0
    \end{smallmatrix}
    from the identidy matrix I3 like this:


    \begin{smallmatrix}
    1&0&0\\ 0&1&0\\ 0&0&1
    \end{smallmatrix}
    Add second row to first row.
    Then substract second row from itself.
    Is the substraction of a row from itself allowed?
     
  2. jcsd
  3. Jun 17, 2014 #2

    Zondrina

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    Homework Helper

    No, you can only add or subtract linear combinations of other rows that are not the row itself when performing row operations.
     
  4. Jun 17, 2014 #3

    LCKurtz

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    Gold Member

    If subtracting a row from itself was a legitimate row operation, you could row reduce any matrix to a zero matrix. And, presto change-o, every matrix has rank zero.
     
  5. Jun 17, 2014 #4

    HallsofIvy

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    Staff Emeritus
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    In fact, it is NOT possible to get
    [tex]\begin{bmatrix}1& 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1\end{bmatrix}[/tex]
    from the identity matrix by "row operations".

    If it were then by doing the "reverse" row operations to the identity matrix would give the inverse matrix. And this matrix, since it has one row of all 0s, has determinant 0 and is NOT invertible.

    (Every row operation has a reverse- the reverse of "multiply row i by a non-zero number" is "divide row i by the number, the reverse of "swap two rows" is itself, and the reverse of "add x (x non zero) times row j to row i" is "subtract 1/x times row j from row i.)
     
    Last edited: Jun 17, 2014
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