Definition 2

Let $I$ be a finite set, $A$ a commutative ring and $X$ a square matrix of type $(I, I)$ over the ring $A$ (II, § 10, no. 7). The determinant of the endomorphism $u$ of the $A$-module $A^I$, whose matrix with respect to the canonical basis of $A^I$ is $X$, is called the determinant of $X$ and denoted by $\det X$.

If $X = (\xi_{ij})_{(i,j) \in I \times I}$ and $(e_i)_{i \in I}$ is the canonical basis of $A^I$, the endomorphism $u$ is then given by

When $I = \{1, n\} \subset \mathbf{N}$, if we write $x_i = u(e_i)$ for $i \in I$, the determinant of $X$ is then defined in the relation

in other words, $\det X$ is equal to the determinant $\det(x_1, x_2, \ldots, x_n)$ with respect to the canonical basis of $A^n$. Consequently: