I have a $2N \times 2N$ matrix called $M$, and then I am interested in the quantity
$$ Tr_{N} Tr_{2} \ln M $$ where the two $Tr$ symbols define partial trace. To be more clear on what I mean, $$Tr _{2} \ln M$$ is a $N\times N$ matrix. The trace is the sum of the two $N\times N$ blocks on the diagonal, basically.
Now, can I use the identity $Tr \ln M = \ln \det M$ to write the first equation as $$ Tr_{N} \ln \det_2 M $$ where $\det_2$ means determinant with respect to the $N\times N$ blocks. Hence, it is like a determinant of a $2\times 2$ matrix, but with block matrices as entries.
I think it is important to specify that these are matrices with complex entries.