Determinant as signed volume

Summary

Let $n \ge 1$ and let $v_1, \dots, v_n \in \R^n$ be the columns of a real $n \times n$ matrix $A$. The parallelepiped spanned by them is $P = \{t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1\}$. Its $n$-dimensional volume is $\abs{\det A}$, where $\det$ is the determinant. If $\det A \ne 0$, then its sign is positive when $(v_1, \dots, v_n)$ has the orientation of the standard basis, and negative when it has the opposite orientation.

Remarks

In $\R^2$, $\abs{ad - bc}$ is the area of the parallelogram spanned by $(a, c)$ and $(b, d)$. For each region $S$ of finite volume, the linear map $x \mapsto Ax$ multiplies the volume of $S$ by $\abs{\det A}$. This fact is the origin of the Jacobian determinant in the change of variables formula.

Explanations

L1 human

The determinant measures how a matrix changes volume. The unit cube goes to a box spanned by the columns, and $\abs{\det A}$ is the volume of this box. A negative determinant means that the matrix also turns space inside out, like a mirror.