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 ^_^