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 $detA=det{A}^T$, 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 byCarl Gustav Jacob Jacobi (1804-1851) [2] in 1827.

2. The $2\times 2$ matrix shows that Jacobi’s theorem does not hold for $2\times 2$matrices. The determinant of the $2n\times 2n$ block matrix withthese $2\times 2$ matrices on the diagonal equals ${(-1)}^n$. Thus Jacobi’s theoremdoes not hold for matrices of even order.

3. For $n=3$, any antisymmetric matrix $A$ can be writtenas

   for some real $v_1,v_2,v_3$, which can be written as avector $v=(v_1,v_2,v_3)$. Then $A$ is the matrix representing themapping $u\mapsto v\times u$, that is, the cross product withrespect to $v$. Since $Av=v\times v=0$, we have $detA=0$.