Example 5

Tag 083D, permanent. Cite it as [Bourbaki, Tag 083D].

Functions of a Real Variable, chapter I, § 1. FIRST DERIVATIVE

The determinant $\det(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n)$ of $n$ vectors $\mathbf{x}_i = (x_{ij})_{1 \leq i \leq n}$ from the space $\mathbf{R}^n$ (Alg., III, p. 522) being a (continuous) multilinear function of the $\mathbf{x}_i$ , one sees that if the $n^2$ real functions $f_{ij}$ are differentiable at $x_0$ , then their determinant $g(x) = \det(f_{ij}(x))$ has a derivative equal to

\sum_{i=1}^n \left[ \mathbf{f}_1(x_0), \ldots, \mathbf{f}_{i-1}(x_0), \mathbf{f}_i'(x_0), \mathbf{f}_{i+1}(x_0), \ldots, \mathbf{f}_n(x_0) \right]

x_0

, where

\mathbf{f}_i(x) = (f_{ij}(x))_{1 \leq j \leq n}

; in other words, one obtains the derivative of a determinant of order

n

by taking the sum of the

n

determinants formed by replacing, for each

i

, the terms of the

i^{th}

column by their derivatives.

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

Cited by

Nothing.

Cites

Nothing.