Are there any useful identities which would help me to find a general formula for
$ \operatorname{Tr} \ln ( I + A ) $
Where I is the identity matrix and A is some N by N complex and symmetric matrix, which has zeroes along the diagonal. I have tried splitting A up into upper and lower triangular components and expanding the log with the mercator series but it's starting to get pretty messy already so I think I missed something. Another property of A which could be useful is that it has a special repeating form along the rows and columns:
\begin{array}{ccccccc} 0 & b & c & d & e ..\\ b & 0 & b & c & d.. \\ c & b & 0 & b & c...\\ d & c & b & 0 & b...\\ . & . & . & . & ...\ \end{array}
It's been a while since I studied matrices so if there is a special name for this I've forgotten!
EDIT: Did a bit of research, this type of matrix is a "Symmetric Toeplitz Matrix".