Lösungen/Solutions/F1S2Ue/Ex6

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Remove["Global`"]

Aufgabe/ Problème 1

M = {{1, 2, 3}, {2, 5, 10}, {4, -1, 41}};  
    MatrixForm[M]

Det[M]

T = {{1, 2, 4}, {2, 5, -1}, {3, 10, 41}};

H = Transpose[T];  MatrixForm[H]

M1 = 
        Transpose[{{1, 0, 0}, {2, 5, -1}, {3, 10, 
              41}}];
M2 = 
        Transpose[{{0, 1, 0}, {2, 5, -1}, {3, 10, 
              41}}];
M3 = 
        Transpose[{{0, 0, 1}, {2, 5, -1}, {3, 10, 41}}];

1* MatrixForm[M1]  +  2* MatrixForm[M2] +  
      4* MatrixForm[M3]

M11 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {3, 10, 
              41}}];
M12 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {3, 10, 
              41}}];
M13 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {3, 10, 
              41}}];
M21 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {3, 10, 
              41}}];
M22 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {3, 10, 
              41}}];
M23 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {3, 10, 
              41}}];
M31 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {3, 10, 
              41}}];
M32 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {3, 10, 
              41}}];
M33 = Transpose[{{0, 0, 1}, {0, 0, 1}, {3, 10, 41}}];

1* (2* MatrixForm[M11]  +  
            5* MatrixForm[M12] +  (-1)* 
              MatrixForm[M13])
 + 
      2* (2* MatrixForm[M21]  +  
            5* MatrixForm[M22] +  (-1)* 
              MatrixForm[M23])
 + 
      4* (2* MatrixForm[M31]  +  
            5* MatrixForm[M32] +  (-1)* 
              MatrixForm[M33])

M111 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {1, 0, 
              0}}];
M121 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {1, 0, 
              0}}];
M131 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {1, 0, 
              0}}];
M211 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {1, 0, 
              0}}];
M221 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {1, 0, 
              0}}];
M231 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {1, 0, 
              0}}];
M311 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {1, 0, 
              0}}];
M321 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {1, 0, 
              0}}];
M331 = Transpose[{{0, 0, 1}, {0, 0, 1}, {1, 0, 0}}];

M112 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {0, 1, 
              0}}];
M122 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {0, 1, 
              0}}];
M132 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {0, 1, 
              0}}];
M212 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {0, 1, 
              0}}];
M222 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {0, 1, 
              0}}];
M232 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {0, 1, 
              0}}];
M312 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {0, 1, 
              0}}];
M322 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {0, 1, 
              0}}];
M332 = Transpose[{{0, 0, 1}, {0, 0, 1}, {0, 1, 0}}];

M113 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {0, 0, 
              1}}];
M123 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {0, 0, 
              1}}];
M133 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {0, 0, 
              1}}];
M213 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {0, 0, 
              1}}];
M223 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {0, 0, 
              1}}];
M233 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {0, 0, 
              1}}];
M313 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {0, 0, 
              1}}];
M323 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {0, 0, 
              1}}];
M333 = Transpose[{{0, 0, 1}, {0, 0, 1}, {0, 0, 1}}];

1* (2* (3*MatrixForm[M111]  + 10* MatrixForm[M112]  +  
                  41 *MatrixForm[M113])
  +  
            5* (3*MatrixForm[M121]  + 10* MatrixForm[M122]  +  
                  41 *MatrixForm[
                      M123]) +  
(-1)* (3*
                    MatrixForm[M131]  + 10* MatrixForm[M132]  +  
                  41 *MatrixForm[M133]))
 + 
      2* (2*  (3*MatrixForm[M211]  + 10* MatrixForm[M212]  +  
                  41 *MatrixForm[M213])  + 
 
            5* (3*MatrixForm[M221]  + 10* MatrixForm[M222]  +  
                  41 *MatrixForm[
                      M223]) + 
 (-1)* (3*
                    MatrixForm[M231]  + 10* MatrixForm[M232]  +  
                  41 *MatrixForm[M233]))
 + 
      4* (2*  (3*MatrixForm[M311]  + 10* MatrixForm[M312]  +  
                  41 *MatrixForm[
                      M313])  +  
5*  (3*
                    MatrixForm[M321]  + 10* MatrixForm[M322]  +  
                  41 *MatrixForm[
                      M323]) +  
(-1)* (3*
                    MatrixForm[M331]  + 10* MatrixForm[M332]  +  
                  41 *MatrixForm[M333]))

(2* (3*MatrixForm[M111]  + 10* MatrixForm[M112]  +  
                  41 *MatrixForm[M113])
  +  
            5* (3*MatrixForm[M121]  + 10* MatrixForm[M122]  +  
                  41 *MatrixForm[
                      M123]) +  
(-1)* (3*
                    MatrixForm[M131]  + 10* MatrixForm[M132]  +  
                  41 *MatrixForm[M133]))
 + 
        2* (2*  (3*MatrixForm[M211]  + 10* MatrixForm[M212]  +  
                    41 *MatrixForm[M213])  + 
 
              5* (3*MatrixForm[M221]  + 10* MatrixForm[M222]  +  
                    41 *MatrixForm[
                        M223]) + 
 (-1)* (3*
                      MatrixForm[M231]  + 10* MatrixForm[M232]  +  
                    41 *MatrixForm[M233]))
 + 
        4* (2*  (3*MatrixForm[M311]  + 10* MatrixForm[M312]  +  
                    41 *MatrixForm[
                        M313])  +  
5*  (3*
                      MatrixForm[M321]  + 10* MatrixForm[M322]  +  
                    41 *MatrixForm[
                        M323]) +  
(-1)* (3*
                      MatrixForm[M331]  + 10* MatrixForm[M332]  +  
                    41 *MatrixForm[M333])) /. 
      MatrixForm -> Det

Aufgabe/ Problème 2

Remove[a, b, c, d, e, f, g, h, i]

M = {{a, b, c}, {d, e, f}, {g, h, i}};  MatrixForm[M]

Det[M]

T = {{a, d, g}, {b, e, h}, {c, f, i}};

H = Transpose[T];  MatrixForm[H]

M1 = 
        Transpose[{{1, 0, 0}, {b, e, h}, {c, f, 
              i}}];
M2 = 
        Transpose[{{0, 1, 0}, {b, e, h}, {c, f, 
              i}}];
M3 = Transpose[{{0, 0, 1}, {b, e, h}, {c, f, i}}];

a* MatrixForm[M1]  +  d* MatrixForm[M2] +  
      g* MatrixForm[M3]

M11 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {c, f, 
              i}}];
M12 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {c, f, 
              i}}];
M13 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {c, f, 
              i}}];
M21 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {c, f, 
              i}}];
M22 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {c, f, 
              i}}];
M23 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {c, f, 
              i}}];
M31 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {c, f, 
              i}}];
M32 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {c, f, 
              i}}];
M33 = Transpose[{{0, 0, 1}, {0, 0, 1}, {c, f, i}}];

a* (b* MatrixForm[M11]  +  e* MatrixForm[M12] +  
            h* MatrixForm[M13])
 + 
      d* (b* MatrixForm[M21]  +  e* MatrixForm[M22] +  
            h* MatrixForm[M23])
 + 
      g* (b* MatrixForm[M31]  +  e* MatrixForm[M32] +  
            h* MatrixForm[M33])

M111 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {1, 0, 
              0}}];
M121 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {1, 0, 
              0}}];
M131 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {1, 0, 
              0}}];
M211 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {1, 0, 
              0}}];
M221 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {1, 0, 
              0}}];
M231 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {1, 0, 
              0}}];
M311 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {1, 0, 
              0}}];
M321 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {1, 0, 
              0}}];
M331 = Transpose[{{0, 0, 1}, {0, 0, 1}, {1, 0, 0}}];

M112 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {0, 1, 
              0}}];
M122 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {0, 1, 
              0}}];
M132 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {0, 1, 
              0}}];
M212 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {0, 1, 
              0}}];
M222 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {0, 1, 
              0}}];
M232 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {0, 1, 
              0}}];
M312 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {0, 1, 
              0}}];
M322 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {0, 1, 
              0}}];
M332 = Transpose[{{0, 0, 1}, {0, 0, 1}, {0, 1, 0}}];

M113 = 
        Transpose[{{1, 0, 0}, {1, 0, 0}, {0, 0, 
              1}}];
M123 = 
        Transpose[{{1, 0, 0}, {0, 1, 0}, {0, 0, 
              1}}];
M133 = 
        Transpose[{{1, 0, 0}, {0, 0, 1}, {0, 0, 
              1}}];
M213 = 
        Transpose[{{0, 1, 0}, {1, 0, 0}, {0, 0, 
              1}}];
M223 = 
        Transpose[{{0, 1, 0}, {0, 1, 0}, {0, 0, 
              1}}];
M233 = 
        Transpose[{{0, 1, 0}, {0, 0, 1}, {0, 0, 
              1}}];
M313 = 
        Transpose[{{0, 0, 1}, {1, 0, 0}, {0, 0, 
              1}}];
M323 = 
        Transpose[{{0, 0, 1}, {0, 1, 0}, {0, 0, 
              1}}];
M333 = Transpose[{{0, 0, 1}, {0, 0, 1}, {0, 0, 1}}];

a* (b* (c*MatrixForm[M111]  + f* MatrixForm[M112]  +  
                  i *MatrixForm[M113])
  +  
            e* (c*MatrixForm[M121]  + f* MatrixForm[M122]  +  
                  i *MatrixForm[M123]) +  
h* (c*
                    MatrixForm[M131]  + f* MatrixForm[M132]  +  
                  i *MatrixForm[M133]))
 + 
      d* (b*  (c*MatrixForm[M211]  + f* MatrixForm[M212]  +  
                  i *MatrixForm[M213])  + 
e* (c*
                    MatrixForm[M221]  + f* MatrixForm[M222]  +  
                  i *MatrixForm[M223]) + 
h* (c*
                    MatrixForm[M231]  + f* MatrixForm[M232]  +  
                  i *MatrixForm[M233]))
 + 
      g* (b*  (c*MatrixForm[M311]  + f* MatrixForm[M312]  +  
                  i *MatrixForm[M313])  +  
e*  (c*
                    MatrixForm[M321]  + f* MatrixForm[M322]  +  
                  i *MatrixForm[M323]) +  
h* (c*
                    MatrixForm[M331]  + f* MatrixForm[M332]  +  
                  i*MatrixForm[M333]))

a* (b* (c*MatrixForm[M111]  + f* MatrixForm[M112]  +  
                      i *MatrixForm[M113])
  +  
                e* (c*MatrixForm[M121]  + f* MatrixForm[M122]  +  
                      i *MatrixForm[M123]) +  
h* (c*
                        MatrixForm[M131]  + f* MatrixForm[M132]  +  
                      i *MatrixForm[M133]))
 + 
          d* (b*  (c*MatrixForm[M211]  + f* MatrixForm[M212]  +  
                      i *MatrixForm[M213])  + 
e* (c*
                        MatrixForm[M221]  + f* MatrixForm[M222]  +  
                      i *MatrixForm[M223]) + 
h* (c*
                        MatrixForm[M231]  + f* MatrixForm[M232]  +  
                      i *MatrixForm[M233]))
 + 
          g* (b*  (c*MatrixForm[M311]  + f* MatrixForm[M312]  +  
                      i *MatrixForm[
                          M313])  +  
e*  (c*
                        MatrixForm[M321]  + f* MatrixForm[M322]  +  
                      i *MatrixForm[M323]) +  
h* (c*
                        MatrixForm[M331]  + f* MatrixForm[M332]  +  
                      i*MatrixForm[M333])) /. 
        MatrixForm -> Det  //  Expand

Det[M]