Consider a $tn\times tn$ matrix $M$. We can see this matrix as a n $n\times n$ matrix of $t\times t$ blocks. It is known that if all blocks commute, than you can compute the determinant $D$ of $M$ in the ring of matrices $t\times t$ using Lagrange's formula. The determinant in the base field of $D$ is then identical to the determinant in the base field of $M$.
Now, I think that one can use Laplace's expansion in this situation if at every step of the expansion the current entry commutes with all blocks of the associated minor. I also think that this may be proved using Dieudonné's notion of determinant for skew fields.
I was wondering, however, if there is a more elementary reference in the literature.
Surprisingly, a result very similar to what I described it has been published this very year:
http://www.sciencedirect.com/science/article/pii/S0024379516304645
It is possible, of course, that the author rediscovered a previously known result.