Elementary Row Operations - only need two?

[clg211]

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.

 

[Hurkyl]

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
$$\begin{array}{cc}<br /> 0 & 1 \\<br /> 1 & 0<br /> \end{array}$$

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