Suppose you weren't allowed to switch rows, would it then always be possible to turn a regular matrix into the unit matrix or would the operation be needed in some cases?
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let's say we want to switch row j and row k. add row j to row k. then add the new row k back to row j.
now multiply row k by -1, and then add row j to row k again. you should now have what row j originally was in the k-th row (and twice the original row j + the original row k in the j-th row).
now subtract twice the current k-th row from the current j-th row. voila! rows j and k have been switched.
(there may be a shorter way to do this, don't know, don't care).
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