If I add a multiple of one row of a square matrix to another row, which is the most common elementary row operation, I don't change the determinant thereby. I know two proofs of this and I have no problem with it. But what does it mean geometrically? I'm going for a more intuitive understanding.
I generated a random 2x2 matrix
then added the first row to the second to get
Then I modeled these two pairs of vectors, and the determinant-parallelograms they generate, on a Geogebra screen (see attachments). I know the two parallelograms should have the same area, and they do, but it's not obvious why. The change in one vector is "undone" by the change in the other, it seems.
I feel I must've missed something. I hope somebody can suggest something.