Permutationen

Needs["DiscreteMath`Permutations`"]

S3 = Permutations[{1, 2, 3}]

Length[S3]

sig3 = Transpose[{Map[Signature, S3]}]

S4 = Permutations[{1, 2, 3, 4}]

Length[S4]

sig4 = Transpose[{Map[Signature, S4]}]

Entwicklungssatz für Determinanten

Untermatrix[A_, {p_, q_}] := 
      Delete[Transpose[Delete[Transpose[A], q]], p]

    
Clear[Determinante]
Determinante[{{a_}}] := a;
Determinante[
        A_] := 
(∑_(k = 1)^(Length[A]))(((-1))^((1 + k))) A[[k, 1]]*
          Determinante[Untermatrix[A, {k, 1}]]

A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}

Untermatrix[A, {2, 3}]

Determinante[A]

A = {{1, 2, 3}, {4, 5, 6}, {7, 8, -8}}

{Det[A], Determinante[A]}