Suppose we are considering a matrix $A\in\Bbb R^{2\times 2}$ withlinearly independent columns. Its determinant $det A$ is the area of the parallelogram enclosed by the two vectors.
Suppose I exchange the rows of the matrix. Then it's determinant becomes $\det A_{transformed}=-\det A$. What is the geometric interpretation of this? I was trying to figure it out by considering that the new parallelogram is flipped along the $x=y$ line, but couldn't go much beyond that.
Can we justify it by saying that flipping the parallelogram along $x=y$ also causes its area vector to flip into the opposite direction?