Jacobi’s theorem

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

Proof. Suppose $A$ is an $n\times n$ square matrix. For the determinant, we then have $\det A=\det A^{T}$ , and $\det(-A)=(-1)^{n}\det A$ . Thus, since $n$ is odd, and $A^{T}=-A$ , we have $\det A=-\det A$ , and the theorem follows. $\Box$

0.0.1 Remarks

1.

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

The $2\times 2$ matrix $\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right)$ shows that Jacobi’s theorem does not hold for $2\times 2$ matrices. The determinant of the $2n\times 2n$ block matrix with these $2\times 2$ matrices on the diagonal equals $(-1)^{n}$ . Thus Jacobi’s theorem does not hold for matrices of even order.
3.

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

$A=\begin{pmatrix}0&-v_{3}&v_{2}\\ v_{3}&0&-v_{1}\\ -v_{2}&v_{1}&0\end{pmatrix}$

for some real $v_{1},v_{2},v_{3}$ , which can be written as a vector $v=(v_{1},v_{2},v_{3})$ . Then $A$ is the matrix representing the mapping $u\mapsto v\times u$ , that is, the cross product with respect to $v$ . Since $Av=v\times v=0$ , we have $\det A=0$ .

References

1 H. Eves, Elementary Matrix 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