Corollary 1

Tag 05XD, permanent. Cite it as [Bourbaki, Tag 05XD].

If $X$ is an invertible square matrix of order $n$ over a field $\mathbf{K}$ , there exist three invertible matrices $P, Q, D$ of order $n$ such that $X = PDQ$ , $P$ and $Q$ being products of matrices of the form $B_{ij}(\lambda)$ and $D$ a diagonal matrix of the form

D = \operatorname{diag}(1, 1, \ldots, \delta),

where $\delta \neq 0$ (cf. Exercise 13).

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

Cited by

Example 3, Appendix 2
Proposition 9, § 17

Cites

Exercise 13