Using the multiplicative compound matrix --see definition at Are compound matrices implemented in mathematica?
one may define also additive compound matrices
(*Multiplicative compound matrix*)
MulCMat[A_?MatrixQ, k_Integer] :=
Module[{m, n, p, q}, {m, n} = Dimensions[A];
p = Subsets[Range[1, m], {k}];
q = Subsets[Range[1, n], {k}];
Table[Det[A[[i, j]]], {i, p}, {j, q}]];
A = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}};
Print["Wiki example of multiplicative compound",
MulCMat[A, 2] // MatrixForm]
(*Additive compound matrix*)
AdCMat[A_, k_Integer] :=
Module[{m, Id}, m = Length[A]; Id = IdentityMatrix[m];
Limit[Simplify[MulCMat[Id + t A, k] - MulCMat[Id, k]]/t, t -> 0]];
A = {{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}};
Print["Second additive compound ", AdCMat[A, 2] // MatrixForm]
Unfortunately, my program does not work on the matrix I have, which is
ja = {{-e \[Beta]e - \[Gamma]r - \[Gamma]s - \[CapitalLambda] +
i (-\[Beta] + \[Nu]), -s \[Beta]e - \[Gamma]r, -\[Gamma]r +
s (-\[Beta] + \[Nu])}, {i \[Beta] + e \[Beta]e,
s \[Beta]e - \[Gamma]e - \[CapitalLambda] + i \[Nu],
s \[Beta] + e \[Nu]}, {0,
ei, -\[Gamma] - \[CapitalLambda] + (-1 + 2 i) \[Nu]}};
AdCMat[ja, 2]
How to fix this?