Proposition 7

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

Let $I$ be a finite set and $X = (\xi_{ji})_{(j, i) \in I \times I}$ a square matrix of type (I, I) over a commutative ring $A$ . Then

\det X = \sum_{\sigma \in S_I} \varepsilon_\sigma \left( \prod_{i \in I} \xi_{\sigma(i), i} \right)

where $\sigma$ runs through the group $S_I$ of permutations of $I$ and $\varepsilon_\sigma$ is the signature of $\sigma$ (I, § 5, no. 7).

Attention may be confined to the case where $I = \{1, n\} \subset \mathbf{N}$ and it then suffices to apply formula (12), where $(e_i)_{1 \leq i \leq n}$ is the canonical basis of $A^n$ and the $x_i$ are the columns of $X$ (cf. no. 3, formula (6)).

In particular, for the determinant of a matrix of order 3

X = \begin{pmatrix} \xi_{11} & \xi_{12} & \xi_{13} \\ \xi_{21} & \xi_{22} & \xi_{23} \\ \xi_{31} & \xi_{32} & \xi_{33} \end{pmatrix}

we have

\det(X) = \xi_{11} \xi_{22} \xi_{33} + \xi_{12} \xi_{23} \xi_{31} + \xi_{21} \xi_{32} \xi_{13} - \xi_{13} \xi_{22} \xi_{31} - \xi_{12} \xi_{21} \xi_{33} - \xi_{11} \xi_{23} \xi_{32}.

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

Cited by

§ 17. Reduced Norms and Traces

Cites

I, § 5, no. 7