Adding a multiple of one row to a multiple of another row

Question

Which of the following are elementary row operation?

(A) Multiplying row 2 by any real number a

(B) Interchange row 1 and row 2

(C) Add 2 times of row 1 to 3 times of row 2

My answer is only (B), because (C) is a combination of 2 elementary row operation, but I am unsure as (C) technically doesn't change the solution set of the system just like any other elementary row operation.

Should (C) be included in the answer?

Thank you.

Answer 1

Note that for (C), you are adding a multiple of one row to a $\textit{multiple}$ of another row. What this means is that instead of adding a multiple of row 1 to just row 2, what you are given is that row 2 has already been multiplied by a constant number that is 3, and then is added by 2$\times$row 1. This means that two-row operations are occurring at once which can not be considered as an elementary row operation. Moreover, (A) is indecisive because you are told to multiply row 2 by any real number $a$ which could be zero and violate the elementary row operation.