# [Math] Prove that $x^TAx=0$ when $x\in \mathbb{R}^n$ and $A$ is antisymmetric

linear algebramatrices

$A$ is real antisymmetric matrix which satisfies

$$A^T=-A$$

Prove that

$$x^TAx=0, \quad x \in \mathbb{R}^n$$

Example of real antisymmetric would be:

$$A=\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}$$

But other than this i dont know where to begin with this.

Now if someone can provide some insight on this that would be much appreciated.

Thanks,

Tuki

Hint: use this$x^TAx=(x^TAx)^T=x^TA^Tx=-x^TAx$.
$x^TAx=(x^TAx)^T$ since $x^TAx$ is a scalar.