  Let $\det_2(A)$ be the block-wise determinant (partial determinant). We
consider the condition for completing the determinant $\det(\det_2(A)) =
\det(A),$ and characterize the case for an arbitrary Kronecker product $A$ 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.
