What would you like to understand?

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: Transformation mirrors or reflects space; Basis vector order is inverted

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: Transformation mirrors or reflects space; Basis vector order is inverted