(%i1) "Cornelsen, S. 224, Nr. 4"$


(%i2) "a)"$


(%i3) M: matrix([0.7,0.3,0.4],[0.1,0.5,0.1],[0.2,0.2,0.5]);
                               [ 0.7  0.3  0.4 ]
                               [               ]
(%o3)                          [ 0.1  0.5  0.1 ]
                               [               ]
                               [ 0.2  0.2  0.5 ]

(%i4) a1: matrix([0.43],[0.22],[0.35]);
                                   [ 0.43 ]
                                   [      ]
(%o4)                              [ 0.22 ]
                                   [      ]
                                   [ 0.35 ]

(%i5) "b)"$


(%i6) M . a1;
                                   [ 0.507 ]
                                   [       ]
(%o6)                              [ 0.188 ]
                                   [       ]
                                   [ 0.305 ]

(%i7) M^^3  . a1;
                                  [ 0.54247 ]
                                  [         ]
(%o7)                             [ 0.17008 ]
                                  [         ]
                                  [ 0.28745 ]

(%i8) M^^5  . a1;
                                 [ 0.5469167 ]
                                 [           ]
(%o8)                            [ 0.1672128 ]
                                 [           ]
                                 [ 0.2858705 ]

(%i9) "Vermutung: siehe M^^5 . a1"$


(%i10) "c)"$


(%i11) "siehe Vermutung in b) oder loese M . x = x"$


(%i12) "d)"$


(%i13) M_d: matrix([0.7,0.3,0.2],[0.1,0.5,0.05],[0.2,0.2,0.75]);
                              [ 0.7  0.3  0.2  ]
                              [                ]
(%o13)                        [ 0.1  0.5  0.05 ]
                              [                ]
                              [ 0.2  0.2  0.75 ]

(%i14) M_d^^5  . a1;
                              [ 0.42849173125  ]
                              [                ]
(%o14)                        [ 0.131817065625 ]
                              [                ]
                              [ 0.439691203125 ]

(%i15) M_d^^15  . a1;
                             [ 0.42593388918133 ]
                             [                  ]
(%o15)                       [ 0.12963370610453 ]
                             [                  ]
                             [ 0.44443240471414 ]

(%i16) "e)"$


(%i17) values;
(%o17)                           [M, a1, M_d]

(%i18) invert(M);
        [ 1.916666666666667   - 0.58333333333333  - 1.416666666666667 ]
        [                                                             ]
(%o18)  [       - 0.25               2.25               - 0.25        ]
        [                                                             ]
        [ - 0.66666666666667  - 0.66666666666667   2.666666666666667  ]

(%i19) invert(M) . a1;
                                    [ 0.2 ]
                                    [     ]
(%o19)                              [ 0.3 ]
                                    [     ]
                                    [ 0.5 ]

(%i20) "f)"$


(%i21) M_f: matrix([0.7,0.5],[0.3,0.5]);
                                 [ 0.7  0.5 ]
(%o21)                           [          ]
                                 [ 0.3  0.5 ]

(%i22) a2: matrix([0.6],[0.4]);
                                    [ 0.6 ]
(%o22)                              [     ]
                                    [ 0.4 ]

(%i23) M_f . a2;
                                   [ 0.62 ]
(%o23)                             [      ]
                                   [ 0.38 ]

(%i24) M_f^^3 . a2;
                                  [ 0.6248 ]
(%o24)                            [        ]
                                  [ 0.3752 ]

(%i25) M_f^^5 . a2;
                                 [ 0.624992 ]
(%o25)                           [          ]
                                 [ 0.375008 ]

(%i26) M_f^^15 . a2;
                             [ 0.62499999999918 ]
(%o26)                       [                  ]
                             [ 0.37500000000082 ]

(%i27) "Vermutung langfristig: DD = 62,5%; DB = 37,5%"$