Partial determinants of Kronecker products

Let ${\det }_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $\det(\det_2(A)) = \det(A),$andcharacterizethecaseforanarbitraryKroneckerproduct$A$ of matrices over an arbitrary field. Further insisting that ${\det }_2(AB)={\det }_2(A){\det }_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $\text{Det}$ which satisfies $\text{Det}({\text{Det}}_2(A))=\text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $\text{Det}$ is defined.