Jacobi’s theorem

Jacobi’s Theorem Any skew-symmetric matrix of odd order has determinantMathworldPlanetmath equal to 0.

Proof. Suppose A is an n×n square matrixMathworldPlanetmath. For the determinant, we then have detA=detAT, and det(-A)=(-1)ndetA. Thus, since n is odd, and AT=-A, we have detA=-detA, and the theorem follows.

0.0.1 Remarks

  1. 1.

    According to [1], this theorem was given by Carl Gustav Jacob Jacobi (1804-1851) [2] in 1827.

  2. 2.

    The 2×2 matrix (01-10) shows that Jacobi’s theorem does not hold for 2×2 matrices. The determinant of the 2n×2n block matrixMathworldPlanetmath with these 2×2 matrices on the diagonal equals (-1)n. Thus Jacobi’s theorem does not hold for matrices of even order.

  3. 3.

    For n=3, any antisymmetric matrix A can be written as


    for some real v1,v2,v3, which can be written as a vector v=(v1,v2,v3). Then A is the matrix representing the mapping uv×u, that is, the cross productMathworldPlanetmath with respect to v. Since Av=v×v=0, we have detA=0.


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath Theory, Dover publications, 1980.
  • 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Jacobi.htmlCarl Gustav Jacob Jacobi
Title Jacobi’s theorem
Canonical name JacobisTheorem
Date of creation 2013-03-22 13:33:06
Last modified on 2013-03-22 13:33:06
Owner Koro (127)
Last modified by Koro (127)
Numerical id 13
Author Koro (127)
Entry type Theorem
Classification msc 15-00