Is combination of elementary row operation considered elementary row operation? I came across this question in my homework if $R_1-R_2-R_3$ considered elementary row operation. My opinion is that it should not be an elementary row operation since it contains three rows which violates the rule "Add a multiple of one row to another row" which only contains two rows. I understand that $R_1-R_2-R_3$ is simply the combination of $R_1-R_2 \to R_1$ and $R_1-R_3 \to R_1$. 

I wouldn't consider it an elementary row operation. In much the same way, when we write 'transposition' (for instace), we mean a very specific kind of permutation, and the result of composing two transpositions is not a transposition itself.
