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 adjoint of $\lambda_0 I-A$, $\mathrm{adj}(\lambda_0 I-A)$, in terms of eigenvectors of $A$ for $\lambda_0$ 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 $\mathrm{adj}(A)$ is provided in terms of those of $A$. Finally, a generalization of the eigenvalue-eigenvector identity and two applications of the Thompson-McEnteggert's formula are presented.