Why do row operations, except swapping rows, dont change the determinant? Looking for a geometric interpretation

Question

Since the determinant is a factor by which an area has changed, shouldn't row operations also change the determinant? For example if we multiply a row with a scalar k, shouldn't the factor by which the area changes increase (or decrease)

Answer 1

Assume two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^2$. As you say, the area of the parallelepiped $\mathbf{0},\mathbf{a}, \mathbf{b}, \mathbf{a}+\mathbf{b}$ is the determinant of the matrix formed by $\mathbf{a}, \mathbf{b}$.

(for simplicity, you can fix $\mathbf{a}=(1,0)$, it is just a rotation/dimension fix).

Now, imagine a second parallelepipied composed, in the same way, from vectors $\mathbf{a}, \mathbf{b}+3\mathbf{a}$. This area is the same than previous, also determinant is the same.