Jacobi’s theorem
Jacobi’s Theorem Any skewsymmetric 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 $detA=det{A}^{T}$, and $det(A)={(1)}^{n}detA$. Thus, since $n$ is odd, and ${A}^{T}=A$, we have $detA=detA$, and the theorem follows. $\mathrm{\square}$
0.0.1 Remarks

1.
According to [1], this theorem was given by Carl Gustav Jacob Jacobi (18041851) [2] in 1827.

2.
The $2\times 2$ matrix $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ 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
$$A=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill {v}_{3}\hfill & \hfill {v}_{2}\hfill \\ \hfill {v}_{3}\hfill & \hfill 0\hfill & \hfill {v}_{1}\hfill \\ \hfill {v}_{2}\hfill & \hfill {v}_{1}\hfill & \hfill 0\hfill \end{array}\right)$$ 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 $detA=0$.
References
 1 H. Eves, Elementary Matrix^{} Theory, Dover publications, 1980.
 2 The MacTutor History of Mathematics archive, http://wwwgap.dcs.stand.ac.uk/ history/Mathematicians/Jacobi.htmlCarl Gustav Jacob Jacobi
Title  Jacobi’s theorem 

Canonical name  JacobisTheorem 
Date of creation  20130322 13:33:06 
Last modified on  20130322 13:33:06 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  13 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 1500 