# Homework 2 Solutions
# 1.
> with(LinearAlgebra):
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> RefPkg/InitPkg.mpl"):
> with(RefPkg);

       [E_I, E_II, E_III, MyDet, MyInverse, MySolve, Ref, XRef]

> A:=<<2,1,0>|<-4,0,3>|<1,-2,0>>;

                              [2    -4     1]
                              [             ]
                         A := [1     0    -2]
                              [             ]
                              [0     3     0]

> A^3-A;

                          [-7     34     9]
                          [               ]
                          [-7     -5    24]
                          [               ]
                          [ 6    -33     3]

> -A^2+A-4*IdentityMatrix(3);

                           [-2    1    -9]
                           [             ]
                           [-1    6    -3]
                           [             ]
                           [-3    3     2]

# 2.
> A:=<<1,2>|<1,0>|<-3,-1>>;
> B:=<<1,-4,3>|<6,5,-1>|<-3,-2,-1>|<3,7,2>>;

                              [1    1    -3]
                         A := [            ]
                              [2    0    -1]


                           [ 1     6    -3    3]
                           [                   ]
                      B := [-4     5    -2    7]
                           [                   ]
                           [ 3    -1    -1    2]

# a
> Transpose(A.B);

                             [-12    -1]
                             [         ]
                             [ 14    13]
                             [         ]
                             [ -2    -5]
                             [         ]
                             [  4     4]

# b
# This cannot be done because  A transpose has 2 columns and B transpose
# has 4 rows.
# c
> A.Transpose(A);

                              [11    5]
                              [       ]
                              [ 5    5]

> Transpose(B).B;

                      [ 26    -17      2    -19]
                      [                        ]
                      [-17     62    -27     51]
                      [                        ]
                      [  2    -27     14    -25]
                      [                        ]
                      [-19     51    -25     62]

> Transpose(B).Transpose(A);

                             [-12    -1]
                             [         ]
                             [ 14    13]
                             [         ]
                             [ -2    -5]
                             [         ]
                             [  4     4]

# 3.
> A:=<<0,0,-2>|<-1,4,2>|<-2,4,0>>;

                             [ 0    -1    -2]
                             [              ]
                        A := [ 0     4     4]
                             [              ]
                             [-2     2     0]

> infolevel[MyInverse]:=3;
> B:=MyInverse(A);

                      infolevel[MyInverse] := 3


                   Elem step: exchange rows , 1, 3


                    [-2     2     0]      [0    0    1]
                    [              ]      [           ]
               AA=, [ 0     4     4], T=, [0    1    0]
                    [              ]      [           ]
                    [ 0    -1    -2]      [1    0    0]


                 Elem step: mult row , 1,  by , -1/2


                   [1    -1     0]      [0    0    -1/2]
                   [             ]      [              ]
              AA=, [0     4     4], T=, [0    1     0  ]
                   [             ]      [              ]
                   [0    -1    -2]      [1    0     0  ]


           Elem step: add , 0,  times row , 1,  to row , 2


                   [1    -1     0]      [0    0    -1/2]
                   [             ]      [              ]
              AA=, [0     4     4], T=, [0    1     0  ]
                   [             ]      [              ]
                   [0    -1    -2]      [1    0     0  ]


           Elem step: add , 0,  times row , 1,  to row , 3


                   [1    -1     0]      [0    0    -1/2]
                   [             ]      [              ]
              AA=, [0     4     4], T=, [0    1     0  ]
                   [             ]      [              ]
                   [0    -1    -2]      [1    0     0  ]


                           Finished col , 1


                  Elem step: mult row , 2,  by , 1/4


                  [1    -1     0]      [0     0     -1/2]
                  [             ]      [                ]
             AA=, [0     1     1], T=, [0    1/4     0  ]
                  [             ]      [                ]
                  [0    -1    -2]      [1     0      0  ]


           Elem step: add , 1,  times row , 2,  to row , 1


                  [1     0     1]      [0    1/4    -1/2]
                  [             ]      [                ]
             AA=, [0     1     1], T=, [0    1/4     0  ]
                  [             ]      [                ]
                  [0    -1    -2]      [1     0      0  ]


           Elem step: add , 1,  times row , 2,  to row , 3


                  [1    0     1]      [0    1/4    -1/2]
                  [            ]      [                ]
             AA=, [0    1     1], T=, [0    1/4     0  ]
                  [            ]      [                ]
                  [0    0    -1]      [1    1/4     0  ]


                           Finished col , 2


                  Elem step: mult row , 3,  by , -1


                  [1    0    1]      [ 0    1/4     -1/2]
                  [           ]      [                  ]
             AA=, [0    1    1], T=, [ 0    1/4      0  ]
                  [           ]      [                  ]
                  [0    0    1]      [-1    -1/4     0  ]


           Elem step: add , -1,  times row , 3,  to row , 1


                  [1    0    0]      [ 1    1/2     -1/2]
                  [           ]      [                  ]
             AA=, [0    1    1], T=, [ 0    1/4      0  ]
                  [           ]      [                  ]
                  [0    0    1]      [-1    -1/4     0  ]


           Elem step: add , -1,  times row , 3,  to row , 2


                  [1    0    0]      [ 1    1/2     -1/2]
                  [           ]      [                  ]
             AA=, [0    1    0], T=, [ 1    1/2      0  ]
                  [           ]      [                  ]
                  [0    0    1]      [-1    -1/4     0  ]


                           Finished col , 3


                           [ 1    1/2     -1/2]
                           [                  ]
                      B := [ 1    1/2      0  ]
                           [                  ]
                           [-1    -1/4     0  ]

> A.B;

                            [1    0    0]
                            [           ]
                            [0    1    0]
                            [           ]
                            [0    0    1]

# 4.
# a.
> infolevel[XRef]:=3:
> A:=<<-1,-2,-1>|<1,2,1>|<2,2,-2>>;
> U:=XRef(A,3,'T');
> T;

                              [-1    1     2]
                              [             ]
                         A := [-2    2     2]
                              [             ]
                              [-1    1    -2]


                                  [-1    1     2]
                                  [             ]
                    i=, 1, j=, 1, [ 0    0    -2]
                                  [             ]
                                  [ 0    0    -4]


                              [ 1    0    0]
                              [            ]
                          L=, [-2    1    0]
                              [            ]
                              [-1    0    1]


                                  [-1    1     2]
                                  [             ]
                    i=, 2, j=, 3, [ 0    0    -2]
                                  [             ]
                                  [ 0    0     0]


                             [ 1     0    0]
                             [             ]
                         L=, [-2     1    0]
                             [             ]
                             [ 3    -2    1]


                              [-1    1     2]
                              [             ]
                         U := [ 0    0    -2]
                              [             ]
                              [ 0    0     0]


                           [ 1     0    0]
                           [             ]
                           [-2     1    0]
                           [             ]
                           [ 3    -2    1]

> T.A;
> U;

                           [-1    1     2]
                           [             ]
                           [ 0    0    -2]
                           [             ]
                           [ 0    0     0]


                           [-1    1     2]
                           [             ]
                           [ 0    0    -2]
                           [             ]
                           [ 0    0     0]

# b.
> E_1:=E_III(3,1,2,-2);
> E_2:=E_III(3,1,3,-1);
> E_3:=E_III(3,2,3,-2);

                               [ 1    0    0]
                               [            ]
                        E_1 := [-2    1    0]
                               [            ]
                               [ 0    0    1]


                               [ 1    0    0]
                               [            ]
                        E_2 := [ 0    1    0]
                               [            ]
                               [-1    0    1]


                               [1     0    0]
                               [            ]
                        E_3 := [0     1    0]
                               [            ]
                               [0    -2    1]

> E_3.E_2.E_1.A;

                           [-1    1     2]
                           [             ]
                           [ 0    0    -2]
                           [             ]
                           [ 0    0     0]

# So T = E_3*E_2*E_1.
# c. Ax=b, so T*Ax=T*b and so U*x=T*b. Thus we have the following
# rightside vectors.
> b1:=<<3,5,1>>;
> b2:=<<3,10,12>>;

                                    [3]
                                    [ ]
                              b1 := [5]
                                    [ ]
                                    [1]


                                    [ 3]
                                    [  ]
                              b2 := [10]
                                    [  ]
                                    [12]

> T.b1;

                                 [ 3]
                                 [  ]
                                 [-1]
                                 [  ]
                                 [ 0]

# So Ux=T*b1 implies that -2*x_3=-1 implies that x_3=1/2 and x_2=x_2 and
# x_1=x_2+2*x_3-3=x_2+1-3=x_2-2. Thus the solution is 
# x_3=(1/2), x_2=x_2 and x_1=x_2-2;
# Now for b2, T*b2 is 
> T.b2;

                                 [3]
                                 [ ]
                                 [4]
                                 [ ]
                                 [1]

# Since there is a 1 in the third row of the vector T*b2 then the system
# has no solutions.