Why is the determinant negative for orientation-reversing transformations in 2D?

The determinant is negative because the transformation flips the relative ordering of the basis vectors (e.g.,

\hat{i}

rotating past

\hat{j}

), reversing the handedness of the coordinate system. The absolute value still gives the area scale, but the sign encodes this directional reversal.

Conditions

Visualize the standard counter-clockwise ordering of basis vectors.
Apply a reflection or rotation that swaps this order.
Note that while area magnitude remains positive, the orientation change assigns a negative sign to the determinant.

Flipping a piece of paper reverses the ordering of basis vectors, resulting in a negative determinant despite preserving area size.

Believing negative area exists physically rather than being a signed measure of orientation.
Ignoring the sign when calculating total scaling factors.