Linear Algebra, 5th Edition

4.5* A Characterization of the Determinant

In Sections 4.2 and 4.3, we showed that the determinant possesses a number of properties. In this section, we show that three of these properties completely characterize the determinant; that is, the only function $\delta :{\text{M}}_{n\times n}(F)\to F$ having these three properties is the determinant. This characterization of the determinant is the one used in Section 4.1 to establish the relationship between $\text{det}\left(\begin{array}{r}u\\ v\end{array}\right)$ and the area of the parallelogram determined by u and v. The first of these properties that characterize the determinant is the one described in Theorem 4.3 (p. 212).

Definition.

A function $\delta :{\text{M}}_{n\times n}(F)\to F$ is called an n-linear function if it is a linear function of each row of an $n\times n$ matrix when the remaining $n-1$