  We use the exterior product of double forms to reformulate celebrated
classical results of linear algebra about matrices and bilinear forms namely
the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton
identities and Jacobi's formula for the determinant. This new formalism is then
used to naturally generalize the previous results to higher multilinear forms
namely to double forms.
  In particular, we show that the Cayley-Hamilton theorem once applied to the
second fundamental form of a hypersurface of the Euclidean space is equivalent
to a linearized version of the Gauss-Bonnet theorem, and once its
generalization is applied to the Riemann curvature tensor (seen as a $(2,2)$
double form) is an infinitisimal version of the general Gauss-Bonnet-Chern
theorem. In addition to that, the general Cayley-Hamilton theorems generate
several universal curvature identities. The generalization of the classical
Laplace expansion of the determinant to double forms is shown to lead to new
general Avez type formulas for all Gauss-Bonnet curvatures.
