M is an N by N matrix with coefficients $a_{ij}$, B an N by N matrix with coefficients $b_{ij}$. Both M and B are symmetric matrices. I am trying to write the determinat of a matrix say, M .* B where B is of the same size and .* represents term by term multiplication. If we write the determinant of M as :
$\det(M) = \sum_{\sigma \in S_n} \left( sgn(\sigma) \prod_{i=1}^n a_{i,\sigma_i}\right)$
Is it possible to write the determinant of C = M .* B as :
$\det(C) = \sum_{\sigma \in S_n} \left( sgn(\sigma) \prod_{i=1}^n b_{i,\sigma_i} a_{i,\sigma_i}\right)$
I just realized my problem may be more complicated. I have a matrix M that depends on a paramater $\rho$ with each entry being $exp(-\rho d_{ij})$ and all diagonal elements equal to 1. So the determinant can be written as : $det(M(\rho)) = \sum_{\sigma \in S_n} \left( sgn(\sigma) \prod_{i=1}^n exp(-\rho d_{i,\sigma_i})\right)$
So the derivative of the determinant wrt $\rho$ gives
$det'(M(\rho)) = \sum_{\sigma \in S_n} \left( sgn(\sigma) \prod_{i=1}^n \sum_{i=1}^n d_{i,\sigma_i} exp(-\rho d_{i,\sigma_i})\right)$.
Is it possible to express this as the determiannt of some matrix that depends on M ?
Yes, it is fine. To see this, note that $c_{i,j}=a_{i, j}b_{i, j}$ for all $i, j$. Hence $c_{i,\sigma(i)} = a_{i,\sigma(i)} b_{i,\sigma(i)}$ for all $i$ and all $\sigma \in S_n$. Using this and the determinant formula will give you the desired result.