  Effective computation of resultants is a central problem in elimination
theory and polynomial system solving. Commonly, we compute the resultant as a
quotient of determinants of matrices and we say that there exists a
determinantal formula when we can express it as a determinant of a matrix whose
elements are the coefficients of the input polynomials. We study the resultant
in the context of mixed multilinear polynomial systems, that is multilinear
systems with polynomials having different supports, on which determinantal
formulas were not known. We construct determinantal formulas for two kind of
multilinear systems related to the Multiparameter Eigenvalue Problem (MEP):
first, when the polynomials agree in all but one block of variables; second,
when the polynomials are bilinear with different supports, related to a
bipartite graph. We use the Weyman complex to construct Koszul-type
determinantal formulas that generalize Sylvester-type formulas. We can use the
matrices associated to these formulas to solve square systems without computing
the resultant. The combination of the resultant matrices with the eigenvalue
and eigenvector criterion for polynomial systems leads to a new approach for
solving MEP.
