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 $$ \left[ {\begin{array}{cc} d & e & f \\ a & b & c \\ \end{array} } \right] $$