The standard relation between the trace and the determinant of matrices is presented in the MO-Q "[Cycling through the zeta garden][1]" where the log and exp functions allow one to jump between additive and multiplicative operations to relate the det and trace. The relations are easiest to understand with diagonalized matrices, i.e., in an eigenvector rep. 

The determinant and trace for a [Fredholm kernel][2] are defined analogously with the kernel of the Fredholm integral as the analog of a matrix and integration over a continuous variable as the analog of discrete matrix multiplications. 

Both the matrix and Fredholm operators have associated zeta functions, and the identities among the symmetric elementary, complete, and power polynomials, associated with the cycle index polynomials for the symmetric groups, certainly encompass these relations. 

**What are some other generalizations of the trace and determinant (and their reciprocal characterizations)?**

(I'm particularly interested to learn if there are analogs in quandle algebra.)

  [1]: http://mathoverflow.net/questions/111770/cycling-through-the-zeta-garden-zeta-functions-for-graphs-cycle-index-polynomi
  [2]: http://en.wikipedia.org/wiki/Fredholm_theory#Fredholm_determinant