Odd-dimensional skew-symmetric singularity theorem
= Odd-dimensional skew-symmetric singularity theorem
{title2=$\det A=0\quad(n\text{ odd},\ A^\top=-A)$}
An <odd integer> number of rows and columns forces a real <skew-symmetric matrix> to be a <singular matrix>. The <determinant> satisfies $\det A=\det A^\top=\det(-A)=(-1)^n\det A=-\det A$, so it is zero. The argument also works over any <field> of <characteristic> different from two.