A question about Elementary Row Operation: Add a Multiple of a Row to Another Row The task is that I have to prove the following statement, using Linear Algebra arguments:

Given a matrix A, then:
To perform an ERO (Elementary Row Operation) type 3 :
(c * R_i)  + R_k --> R_k  (i.e.  replace a row k by adding c-multiple of row i to row k)
is the same as replacing a row k by subtracting a multiple of some row from another row

I just don't know how to formally prove this statement, like how the arguments should look like.
By some inspections, I'm pretty sure that doing

(c * R_i) + R_k --> R_k

is the same as doing:

R_i - (d * R_k) --> R_k

where d can be positive or negative, but it must have opposite sign with c.  I use an example as follows:
A = (2 1 3, 4 3 1)
Then if I want to add row 2 to row 1, say, instead of doing (1 * 4) + 2 --> 6 and so on, I do 4 - [ (-1) * 2 ] --> 6 instead.  Thus c = 1 and d = -1 in this case.
That's why I conclude that the coefficient d should always be the opposite sign with coefficient c.
Would someone help me on how to construct a formal proof of the statement?  I know how to go about the examples, but I understand examples are never proofs >_<
Thank you very much ^_^

Note: If $c$ is some non-zero scalar, then 


*

*adding $cR_i$ to $R_k$ and replacing the original $R_{k\text{ old}}$ by $(R_k + cR_i)$ 


is the same as  


*

*subtracting $−c⋅R_i$ from $R_k$ and replacing the old $R_k$ by the result $R_k - (-cR_i)$.


Since...$R_k + cR_i = R_k - (-cR_i)$
