PlanetMath: Jacobi's theorem

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Jacobi's theorem

(Theorem)

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

Proof. Suppose 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 is odd, and , we have $\det A = -\det A$ , and the theorem follows. $\Box$

Remarks

According to [1], this theorem was given by Carl Gustav Jacob Jacobi (1804-1851) [2] in 1827.
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 . Thus Jacobi's theorem does not hold for matrices of even order.
For , any antisymmetric matrix can be written as
$\displaystyle A = \begin{pmatrix} 0 & -v_3 & v_2 \ v_3 & 0 & -v_1 \ -v_2 & v_1 & 0 \end{pmatrix}$
for some real , which can be written as a vector . Then is the matrix representing the mapping $u\mapsto v\times u$ , that is, the cross product with respect to . Since $Av=v\times v=0$ , we have $\det A=0$ .