The Elementary Row Operations are stated as follows:
- Interchange two rows.
- Multiply a row by a nonzero constant.
- Add a multiple of a row to another row.
Consider the matrix below:
$$A =\begin{bmatrix}5 & 2 & 0 & 0 & -2\\0 & 1 & 4 & 3 & 2\\0 & 0 & 2 & 6 & 3\\0 & 0 & 3 & 4 & 1\\0 & 0 & 0 & 0 & 2 \end{bmatrix}$$
I am concerned with the third operation which is stated as $r_i \rightarrow r_i+cr_j$. Why is it wrong to interpret it as $r_i \rightarrow cr_i+r_j$? Is there any proof for this? Thank you!