Corollary 3

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

Let $u$ be an endomorphism of a finite-dimensional free $\mathbf{A}$ -module $M$ and $^t u$ the transpose endomorphism of the dual $M^*$ (II, § 2, no. 5, Definition 5); then

\det(^t u) = \det(u).

X

is the matrix of

u

with respect to a basis of

M

^t X

is the matrix of

^t u

with respect to the dual basis (II, § 10, no. 4, Proposition 3); as

\det(u) = \det(X)

and

\det(^t u) = \det(^t X)

, the conclusion follows from Proposition 8.

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

Cited by

Corollary 3, § 17

Cites

II, § 10, no. 4, Proposition 3
II, § 2, no. 5, Definition 5
Proposition 8