The trace formulas yield the inverse metric formula

It is a well-known fact that the first and last nontrivial coefficients of the characteristic polynomial of a linear operator are, respectively, its t race and its determinant. This work shows how to compute recursively all th e coefficients as polynomial functions in the traces of the successive powe rs of the operator. With the aid of Cayley-Hamilton's theorem the trace for mulas provide a rational formula for the resolvent kernel and an operator-v alued null identity for each finite dimension of the underlying vector spac e. The four-dimensional resolvent formula allows an algebraic solution of t he inverse metric problem in general relativity. (C) 1998 American Institut e of Physics. [S0022-2488(98)01311-5].