On a formula of Thompson and McEnteggert for the adjugate matrix

For an eigenvalue λ0 of a Hermitian matrix A, the formula Thompson and McEnteggert gives explicit expression adjugate λ0I−A, Adj(λ0I−A), in terms eigenvectors A for all its eigenvalues. In this paper Thompson-McEnteggert's is generalized to include any with entries arbitrary field. addition, nonsingular elementary divisors Adj(A) provided those A. Finally, generalization eigenvalue-eigenvector identity three applications are presented.