Elementary Row Operations - only need two?

clg211
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I was reading that the "exchange rows" operation can be achieved through the other two operations: multiplication by a nonzero number and adding a multiple of one row to another.

Any thoughts on the actual algorithm for achieving an exchange of rows through these other two operations? I realize it should be an iterative process, but not sure where it would start.
 
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I would start with the elementary matrix for a swap, and try and reduce it to row echelon form using only adds and multiplies.

You start with this
<br /> \begin{array}{cc}<br /> 0 &amp; 1 \\<br /> 1 &amp; 0<br /> \end{array}<br />

Now, where to begin? How about the usual place; we want a 1 in the top-left corner...
 

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