  By using combinatorics, we give a new proof for the recurrence relations of
the characteristic polynomial coefficients, and then we obtain an explicit
expression for the generic term of the coefficient sequence, which yields the
trace formulae of the Cayley-Hamilton's theorem with all coefficients
explicitly given, and which implies a byproduct, a complete expression for the
determinant of any finite-dimensional matrix in terms of the traces of its
successive powers. And we discuss some of their applications to chiral
perturbation theory and general relativity.
