row operations, swapping rows How I can justify that swap rows in an arbitrary matrix, can be done with other operations with rows? This means that a matrix can be reduced without swapping rows. Is this true for any matrix? Thanks for your help.

A row swap operation can be done with the following sequence of the other operations, illustrated for rows one and two, with hopefully obvious notation $$ {r_1\atop r_2} \rightarrow {r_1+r_2\atop r_2}\rightarrow {r_1+r_2\atop r_2-(r_1+r_2)}= {r_1+r_2\atop -r_1}\rightarrow {(r_1+r_2)-r_1\atop-r_1}= {r_2\atop -r_1}\rightarrow {r_2\atop r_1}. $$ The same sequence of operations will swap two distinct rows $r_k$ and $r_j$.