Originally Posted by
NonCommAlg since $\displaystyle \det(-A)=(-1)^n \det A,$ we have $\displaystyle \det(-A)=\det A$ if and only if at least one of theses conditions holds:
1) $\displaystyle \det A = 0,$
2) $\displaystyle 2 \mid n,$
3) $\displaystyle \text{char}(F)=2.$