What does the determinant of a matrix represent geometrically?

The determinant of a matrix has a significant geometric interpretation, particularly in the context of linear transformations. When applied to a geometric shape, such as a rectangle or a parallelogram, the absolute value of the determinant represents the scaling factor by which the transformation alters the area (in two dimensions) or volume (in three dimensions) of that shape.

For a two-dimensional transformation, if you take a rectangle and apply a 2x2 transformation matrix to it, the area of the transformed shape is equal to the area of the original rectangle multiplied by the absolute value of the determinant of that matrix. In three dimensions, this concept extends to volume, indicating how the matrix transformation affects the volume of a solid shape.

Additionally, the sign of the determinant can imply whether the transformation preserves or reverses orientation. A positive determinant indicates that the orientation is preserved, while a negative determinant indicates a reversal. Hence, while the determinant provides information about orientation and scaling, its primary geometric representation is related to the volume (or area in lower dimensions) of the transformed figure.

This means that the correct understanding of the determinant encompasses both its scaling effect and its role in transformations, with the most straightforward interpretation being its volume representation in higher dimensions.