Corollary 1

Tag 064H, permanent. Cite it as [Bourbaki, Tag 064H].

For n vectors $x_1, \ldots, x_n$ of $\mathbf{A}^n$ , let $Y(x_1, \ldots, x_n)$ denote the square matrix of order n whose i-th row is $x_i$ , for $1 \leq i \leq n$ . Then the mapping

(x_1, \ldots, x_n) \mapsto \det(Y(x_1, \ldots, x_n))

from

(\mathbf{A}^n)^n

\mathbf{A}

is alternating and n-linear.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 8. Determinants
Proposition 11, § 8

Cites

Nothing.