On a formula of Thompson and McEnteggert for the adjugate matrix

For an eigenvalue lambda(0) of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of lambda I-0 - A, Adj(lambda I-0 - A), in terms of eigenvectors of Afor lambda(0) and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A, a formula for the elementary divisors of Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented. Partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Partially supported by "Ministerio de Economia, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grants MTM2017-83624-P and MTM2017-90682-REDT, and by UPV/EHU through grant GIU16/42.