  It is a well-known fact that the first and last non-trivial coefficients of
the characteristic polynomial of a linear operator are respectively its trace
and its determinant. This work shows how to compute recursively all the
coefficients as polynomial functions in the traces of successive powers of the
operator. With the aid of Cayley-Hamilton's theorem the trace formulas provide
a rational formula for the resolvent kernel and an operator-valued null
identity for each finite dimension of the underlying vector space. The
4-dimensional resolvent formula allows an algebraic solution of the inverse
metric problem in general relativity.
