# Geometric interpretation of det\det(A^T) = \det(A)

$$\det(A^T) = \det(A)\det(A^T) = \det(A)$$

Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?

A geometric interpretation in four intuitive steps….

The Determinant is the Volume Change Factor

Think of the matrix as a geometric transformation, mapping points (column vectors) to points: $x \mapsto Mx$.
The determinant $\mbox{det}(M)$ gives the factor by which volumes change under this mapping.

For example, in the question you define the determinant as the volume of the parallelepiped whose edges are given by the matrix columns. This is exactly what the unit cube maps to, so again, the determinant is the factor by which the volume changes.

A Matrix Maps a Sphere to an Ellipsoid

Being a linear transformation, a matrix maps a sphere to an ellipsoid.
The singular value decomposition makes this especially clear.

If you consider the principal axes of the ellipsoid (and their preimage in the sphere), the singular value decomposition expresses the matrix as a product of (1) a rotation that aligns the principal axes with the coordinate axes, (2) scalings in the coordinate axis directions to obtain the ellipsoidal shape, and (3) another rotation into the final position.

The Transpose Inverts the Rotation but Keeps the Scaling

The transpose of the matrix is very closely related, since the transpose of a product is the reversed product of the transposes, and the transpose of a rotation is its inverse. In this case, we see that the transpose is given by the inverse of rotation (3), the same scaling (2), and finally the inverse of rotation (1).

(This is almost the same as the inverse of the matrix, except the inverse naturally uses the inverse of the original scaling (2).)

The Transpose has the Same Determinant

Anyway, the rotations don’t change the volume — only the scaling step (2) changes the volume. Since this step is exactly the same for $M$ and $M^\top$, the determinants are the same.