Cramersche Regel

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

Adjunkte[A_, {j_, k_}] := (-1)^(j + k)*
        Det[Untermatrix[A, {j, k}]]

CramerInverse[A_] := 
      Table[Adjunkte[A, {k, j}], {j, Length[A]}, {k, Length[A]}]/
        Det[A]

A = {{6, 5}, {7, 6}}

Inverse[A]

CramerInverse[A]

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

Inverse[A]

CramerInverse[A]

A = Array[a, {2, 2}]

Inverse[A]

CramerInverse[A]

Det[A] Inverse[A]

A = Array[a, {3, 3}]

Det[A] Inverse[A]

Generierung einer Matrix mit Determinante 1

While[Not[Det[A] === 1], 
        A = Table[
            Random[Integer, {-5, 
                5}], {3}, {3}]];
A

Gleichungssystem

system = 
      A . {{x}, {y}, {z}} == 
        Table[{Random[Integer, {-5, 5}]}, {3}]

Solve[system, {x, y, z}]

Eigenwerte

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

charpol = Det[A - λ IdentityMatrix[3]]

CharacteristicPolynomial[A, λ]

Solve[charpol == 0, λ]

Eigenvalues[A]

Beispiel 9.1

A = {{1 - 13  I, 0, 2, 
          7  I}, {3, I, I, 
          4}, {2 - I, 0, (1/7), 1}, {0, 0, I, 
          0}}

charpol = Det[A - λ IdentityMatrix[4]]

Eigenvalues[A]

A = Array[a, {3, 3}]

charpol = Det[A - λ IdentityMatrix[3]]

Eigenvalues[A]