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I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it.

Given some arbitrary matrix, how can two rows be interchanged using only a finite number of the other two elementary row operations (Adding multiple of one row to another and multiplying a row by some constant)?

Let's suppose our arbitrary matrix is the following $$ \left[ {\begin{array}{cc} a & b & c \\ d & e & f \\ \end{array} } \right] $$ What series of elementary row operations (excluding interchange) will result in the matrix $$ \left[ {\begin{array}{cc} d & e & f \\ a & b & c \\ \end{array} } \right] $$

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Hint: start by adding row $1$ to row $2$, then add $-1$ times row $2$ to row $1$....

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  • $\begingroup$ Ah thank you. I wasn't sure how to start out. I appreciate the help. $\endgroup$ Jan 25, 2015 at 23:37
  • $\begingroup$ @RobertIsrael The interchange operation can be written in terms of the other two elementary row operations. How can I conclude that "interchanging two rows can be effected by elementary row operations of the other two types" from this? $\endgroup$
    – zaira
    Jan 15, 2020 at 6:21
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For completion, you can do this in-place (that is, no extra storage):

As individual operations:

  1. Set row 1 to row 1 plus row 2.
  2. Set row 2 to row 2 subtracted from row 1.
  3. Set row 1 to row 2 subtracted from row 1.
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  • $\begingroup$ The interchange operation can be written in terms of the other two elementary row operations. How can I conclude that "interchanging two rows can be effected by elementary row operations of the other two types" from this? $\endgroup$
    – zaira
    Jan 15, 2020 at 6:22

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