  Often in mathematics it is useful to summarize a multivariate phenomenon with
a single number and in fact, the determinant -- which is represented by det --
is one of the simplest cases. In fact, this number it is defined only for
square matrices and a lot of its properties are very well-known. For instance,
the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is
not, in general, an additive function. Another interesting function in the
matrix analysis is the characteristic polynomial -- in fact, given a matrix A,
this function is defined by $p_A(t)=det(tI-A)$ where I is the identity matrix
-- which elements are, up a sign, the elementary symmetric functions associated
to the eigenvalues of the matrix A. In the present paper new expressions
related with the determinant of sum of matrices and the elementary symmetric
functions are given. Moreover, the connection with the Mobius function and the
partial ordered sets (poset) is presented. Finally, a problem related with the
determinant of sum of matrices is solved.
