Do all matrices appearing in Gauss–Jordan elimination represent equivalent linear transformations?

Question

There are three types of elementary row operations in Gauss–Jordan elimination:

1. Swapping two rows,
2. Multiplying a row by a nonzero number,
3. Adding a multiple of one row to another row.

From my understanding Gauss–Jordan elimination works precisely because it doesn't change the underlying linear transformation the original matrix represents.

I guess it's intuitively clear to me how 1 and 2 will not change the underlying linear transformation, but the 3rd one is not so trivial to reason about.

So my question is - am I right that none of the elementary row operations in Gauss–Jordan elimination change the underlying linear transformation? And is there a way to present more intuitively why adding a multiple of one row to another row does not change the linear transformation?