Proposition 11

Let $M$ be a free $A$-module of finite dimension $n$; for every endomorphism $u$ of $M$ and every ordered pair of elements $\xi, \eta$ of $A$,

Let $(e_i)_{1 \leq i \leq n}$ be a basis of $M$ and let $I = \{1, n\}$; to calculate the left hand side of (18), we must form the product

which is equal to the sum of the terms $\xi^{n-p} \eta^p z_K$, where

with $x_i = u(e_i)$ for $i \in K$, $x_i = e_i$ for $i \in H = I - K$, where the integer $p$ runs through the interval $[0, n]$ and, for each $p$, $K$ runs through the set of subsets of I with $p$ elements. If $i_1 < i_2 < \cdots < i_{n-p}$ (resp. $j_1 < j_2 < \cdots < j_p$) are the elements of H (resp. K) arranged in increasing order, we can write ($§ 7$, no. 8, Corollary 1 to Theorem 1 and formula (19))

But if $X$ is the matrix of $u$ with respect to the basis $(e_i)$, then by Proposition 10

and hence

Now $H \cap L \neq \varnothing$ except for $L = K$; it therefore follows from $§ 7$, no. 8, formula (20) that $z_K = (\det X_{K, K}) e_1 \wedge e_2 \wedge \cdots \wedge e_n$ and formula (18) then follows from Proposition 10 and the definition of the trace of a matrix (II, $§ 10$, no. 11, formulae (49) and (50)).