In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{ kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base field is not of characteristic $2$. Considering some symmetries in that formula, we found a new formula so that \begin{ equation*} \operatorname{ Crank}(det_n) \leq \operatorname{ rank}(det_n) \leq \frac{ n!}{ 2^{ \lfloor(n-2)/2 \rfloor}} \end{ equation*} when the base field is not of characteristic $2$.
Categories: math.AC math.RA