Is combination of elementary row operation considered elementary row operation?

Question

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 $R_1-R_3 \to R_1$.

Answer 1

$R_1-R_2-R_3$ is not a row operation, let alone an elementary row operation, but a row vector. Row operations can be described by formulas, e.g., in the following ways: $$R_1\ \rightsquigarrow \ R_1-R_2-R_3\>,\qquad {\rm or}\qquad R_1\ \leftarrow \ R_1-R_2-R_3\ .$$ This said, my answer to your main question is No. Elementary row operations affect only two rows. Of course one can concatenate several elementary row operations into a single step, which is then declared as "using elementary row operations we arrive at $\ldots$" .

Answer 2

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.