If A_{ij} is a m \times m matrix and B_{ij} is a n \times n matrix, the direct product is
C = A \otimes B
where C is an mn \times mn matrix with elements
C_{\alpha \beta} = A_{ij} B_{kl}
with
\alpha = n (i-1) + k , \;\;\;\; \beta = n(j-1) + l.
The determinant of C_{\alpha \beta} is given by the usual formula
<br />
det C_{\alpha \beta} = {1 \over (mn)!} \sum_{\alpha_1 \beta_1} \cdots \sum_{\alpha_{mn} \beta_{mn}} <br />
\epsilon_{\alpha_1 \dots \alpha_{mn}} \epsilon_{\beta_1 \dots \beta_{mn}} <br />
C_{\alpha_1 \beta_1} \dots C_{\alpha_{mn} \beta_{mn}}<br />.
In terms of A_{ij} and B_{kl} this becomes
<br />
det C_{\alpha \beta} = {1 \over (mn)!} \sum_{i_1, j_1, k_1, l_1} \cdots <br />
\sum_{i_{mn}, j_{mn}, k_{mn}, l_{mn}} <br />
\epsilon_{n(i_1 - 1) + k_1 \cdots n(i_{mn} - 1) + k_{mn}} <br />
\epsilon_{n(j_1 - 1) + l_1 \cdots n(j_{mn} - 1) + l_{mn}} <br />
<br />
A_{i_1 j_1} B_{k_1 l_1} \dots A_{i_{mn} j_{mn}} B_{k_{mn} l_{mn}}<br />.
Looks a bit daunting
