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 / g / Solution

The determinant is invariant under transpose, while multiplying all $n$ rows by $-1$ multiplies it by $(-1{)}^n$. Thus, for odd integer $n$,Over the real numbers, this forces $\det A=0$, so $A$ is a singular matrix. This is the odd-dimensional skew-symmetric singularity theorem.