Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 6 / e

If $A$ is a skew-symmetric matrix, the scalar $x^{\top }Ax$ equals its own transpose, soand it is zero over the real numbers.

Conversely, suppose the quadratic form $x^{\top }Ax$ vanishes for every real vector. Substituting $x=e_i$ gives $a_{ii}=0$. Substituting $x=e_i+e_j$ gives $a_{ij}+a_{ji}=0$ for $i\ne j$. Therefore $A^{\top }=-A$. This proves the quadratic-form criterion for a skew-symmetric matrix: