> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> RefPkg/InitPkg.mpl");

  libname := /afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlg\

        ebra/Maple, "/afs/bp/dist/maple60/lib"

> with(LinearAlgebra):
> with(RefPkg);

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

> A := Matrix(4,4);

                            [0    0    0    0]
                            [                ]
                            [0    0    0    0]
                       A := [                ]
                            [0    0    0    0]
                            [                ]
                            [0    0    0    0]

> for i from 1 to 4 do for j from 1 to 4 do A[i,j]:=a[i,j]; od: od:
> A;

              [a[1, 1]    a[1, 2]    a[1, 3]    a[1, 4]]
              [                                        ]
              [a[2, 1]    a[2, 2]    a[2, 3]    a[2, 4]]
              [                                        ]
              [a[3, 1]    a[3, 2]    a[3, 3]    a[3, 4]]
              [                                        ]
              [a[4, 1]    a[4, 2]    a[4, 3]    a[4, 4]]

> E1 := E_I(4,2,3);

                             [1    0    0    0]
                             [                ]
                             [0    0    1    0]
                       E1 := [                ]
                             [0    1    0    0]
                             [                ]
                             [0    0    0    1]

> E1 . A;

              [a[1, 1]    a[1, 2]    a[1, 3]    a[1, 4]]
              [                                        ]
              [a[3, 1]    a[3, 2]    a[3, 3]    a[3, 4]]
              [                                        ]
              [a[2, 1]    a[2, 2]    a[2, 3]    a[2, 4]]
              [                                        ]
              [a[4, 1]    a[4, 2]    a[4, 3]    a[4, 4]]

> A . E1;

              [a[1, 1]    a[1, 3]    a[1, 2]    a[1, 4]]
              [                                        ]
              [a[2, 1]    a[2, 3]    a[2, 2]    a[2, 4]]
              [                                        ]
              [a[3, 1]    a[3, 3]    a[3, 2]    a[3, 4]]
              [                                        ]
              [a[4, 1]    a[4, 3]    a[4, 2]    a[4, 4]]

> MatrixInverse(E1);

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

> E2 := E_II(4,2,alpha);

                           [1      0      0    0]
                           [                    ]
                           [0    alpha    0    0]
                     E2 := [                    ]
                           [0      0      1    0]
                           [                    ]
                           [0      0      0    1]

> E2 . A;

        [a[1, 1] , a[1, 2] , a[1, 3] , a[1, 4]]

        [alpha a[2, 1] , alpha a[2, 2] , alpha a[2, 3] ,

        alpha a[2, 4]]

        [a[3, 1] , a[3, 2] , a[3, 3] , a[3, 4]]

        [a[4, 1] , a[4, 2] , a[4, 3] , a[4, 4]]

> A . E2;

           [a[1, 1]    a[1, 2] alpha    a[1, 3]    a[1, 4]]
           [                                              ]
           [a[2, 1]    alpha a[2, 2]    a[2, 3]    a[2, 4]]
           [                                              ]
           [a[3, 1]    a[3, 2] alpha    a[3, 3]    a[3, 4]]
           [                                              ]
           [a[4, 1]    a[4, 2] alpha    a[4, 3]    a[4, 4]]

> MatrixInverse(E2);

                        [1      0      0    0]
                        [                    ]
                        [       1            ]
                        [0    -----    0    0]
                        [     alpha          ]
                        [                    ]
                        [0      0      1    0]
                        [                    ]
                        [0      0      0    1]

> E3 := E_III(4,3,4,alpha);

                           [1    0      0      0]
                           [                    ]
                           [0    1      0      0]
                     E3 := [                    ]
                           [0    0      1      0]
                           [                    ]
                           [0    0    alpha    1]

> E3 . A;

        [a[1, 1] , a[1, 2] , a[1, 3] , a[1, 4]]

        [a[2, 1] , a[2, 2] , a[2, 3] , a[2, 4]]

        [a[3, 1] , a[3, 2] , a[3, 3] , a[3, 4]]

        [alpha a[3, 1] + a[4, 1] , a[3, 2] alpha + a[4, 2] ,

        alpha a[3, 3] + a[4, 3] , alpha a[3, 4] + a[4, 4]]

> A . E3;

       [a[1, 1] , a[1, 2] , a[1, 3] + a[1, 4] alpha , a[1, 4]]
       [                                                     ]
       [a[2, 1] , a[2, 2] , a[2, 3] + alpha a[2, 4] , a[2, 4]]
       [                                                     ]
       [a[3, 1] , a[3, 2] , a[3, 3] + alpha a[3, 4] , a[3, 4]]
       [                                                     ]
       [a[4, 1] , a[4, 2] , a[4, 3] + a[4, 4] alpha , a[4, 4]]

> MatrixInverse(E3);

                       [1    0      0       0]
                       [                     ]
                       [0    1      0       0]
                       [                     ]
                       [0    0      1       0]
                       [                     ]
                       [0    0    -alpha    1]

>