> 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(RefPkg);

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

> ?XRef
> 
# Examples:
> infolevel['RefPkg']:=1:
> with(RefPkg, XRef);

                                [XRef]

> A := Matrix([ [1, 2, 1, 3], [3, -1, -3, -1], [2, 3, 1, 4] ]);

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

> U := XRef(A, 3,'T','L');

                         [1     2     1       3  ]
                         [                       ]
                    U := [0    -7     -6     -10 ]
                         [                       ]
                         [0     0    -1/7    -4/7]

> T;

                         [  1       0      0]
                         [                  ]
                         [ -3       1      0]
                         [                  ]
                         [-11/7    -1/7    1]

> T . A;

                      [1     2     1       3  ]
                      [                       ]
                      [0    -7     -6     -10 ]
                      [                       ]
                      [0     0    -1/7    -4/7]

> L;

                           [1     0     0]
                           [             ]
                           [3     1     0]
                           [             ]
                           [2    1/7    1]

> T . L;

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

> L . U;

                        [1     2     1     3]
                        [                   ]
                        [3    -1    -3    -1]
                        [                   ]
                        [2     3     1     4]

> B := Matrix(2,2,[[4,1],[-2,3]]);

                                 [ 4    1]
                            B := [       ]
                                 [-2    3]

> with(LinearAlgebra):
> Determinant(B);

                                  14

> MatrixInverse(B);

                            [        -1 ]
                            [3/14    -- ]
                            [        14 ]
                            [           ]
                            [1/7     2/7]

> C := Matrix(3,3);

                               [0    0    0]
                               [           ]
                          C := [0    0    0]
                               [           ]
                               [0    0    0]

> for i from 1 to 3 do for j from 1 to 3 do C[i,j]:=c[i,j]: od: od:
> C;

                   [c[1, 1]    c[1, 2]    c[1, 3]]
                   [                             ]
                   [c[2, 1]    c[2, 2]    c[2, 3]]
                   [                             ]
                   [c[3, 1]    c[3, 2]    c[3, 3]]

> MatrixInverse(C);

        [c[2, 2] c[3, 3] - c[2, 3] c[3, 2]
        [--------------------------------- ,
        [               %1

          c[1, 2] c[3, 3] - c[1, 3] c[3, 2]
        - --------------------------------- ,
                         %1

          -c[1, 2] c[2, 3] + c[1, 3] c[2, 2]]
        - ----------------------------------]
                          %1                ]

        [  -c[3, 1] c[2, 3] + c[2, 1] c[3, 3]
        [- ---------------------------------- ,
        [                  %1

        -c[3, 1] c[1, 3] + c[1, 1] c[3, 3]
        ---------------------------------- ,
                        %1

          -c[2, 1] c[1, 3] + c[1, 1] c[2, 3]]
        - ----------------------------------]
                          %1                ]

        [  c[3, 1] c[2, 2] - c[2, 1] c[3, 2]
        [- --------------------------------- ,
        [                 %1

        c[3, 1] c[1, 2] - c[1, 1] c[3, 2]
        --------------------------------- ,
                       %1

        -c[2, 1] c[1, 2] + c[1, 1] c[2, 2]]
        ----------------------------------]
                        %1                ]

  %1 := c[3, 1] c[1, 2] c[2, 3] - c[3, 1] c[1, 3] c[2, 2]

         - c[2, 1] c[1, 2] c[3, 3] + c[2, 1] c[1, 3] c[3, 2]

         + c[1, 1] c[2, 2] c[3, 3] - c[1, 1] c[2, 3] c[3, 2]

> Determinant(C);

  c[3, 1] c[1, 2] c[2, 3] - c[3, 1] c[1, 3] c[2, 2]

         - c[2, 1] c[1, 2] c[3, 3] + c[2, 1] c[1, 3] c[3, 2]

         + c[1, 1] c[2, 2] c[3, 3] - c[1, 1] c[2, 3] c[3, 2]

> unprotect(D);
> D := Matrix(4,4);

                            [0    0    0    0]
                            [                ]
                            [0    0    0    0]
                       D := [                ]
                            [0    0    0    0]
                            [                ]
                            [0    0    0    0]

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

              [d[1, 1]    d[1, 2]    d[1, 3]    d[1, 4]]
              [                                        ]
              [d[2, 1]    d[2, 2]    d[2, 3]    d[2, 4]]
              [                                        ]
              [d[3, 1]    d[3, 2]    d[3, 3]    d[3, 4]]
              [                                        ]
              [d[4, 1]    d[4, 2]    d[4, 3]    d[4, 4]]

> Determinant(D);

  d[1, 1] d[2, 2] d[3, 3] d[4, 4] - d[1, 1] d[2, 2] d[3, 4] d[4, 3]

         + d[1, 1] d[3, 2] d[4, 3] d[2, 4]

         - d[1, 1] d[3, 2] d[2, 3] d[4, 4]

         + d[1, 1] d[4, 2] d[2, 3] d[3, 4]

         - d[1, 1] d[4, 2] d[3, 3] d[2, 4]

         - d[2, 1] d[1, 2] d[3, 3] d[4, 4]

         + d[2, 1] d[1, 2] d[3, 4] d[4, 3]

         - d[2, 1] d[3, 2] d[4, 3] d[1, 4]

         + d[2, 1] d[3, 2] d[1, 3] d[4, 4]

         - d[2, 1] d[4, 2] d[1, 3] d[3, 4]

         + d[2, 1] d[4, 2] d[3, 3] d[1, 4]

         + d[3, 1] d[1, 2] d[2, 3] d[4, 4]

         - d[3, 1] d[1, 2] d[4, 3] d[2, 4]

         + d[3, 1] d[2, 2] d[4, 3] d[1, 4]

         - d[3, 1] d[2, 2] d[1, 3] d[4, 4]

         + d[3, 1] d[4, 2] d[1, 3] d[2, 4]

         - d[3, 1] d[4, 2] d[2, 3] d[1, 4]

         - d[4, 1] d[1, 2] d[2, 3] d[3, 4]

         + d[4, 1] d[1, 2] d[3, 3] d[2, 4]

         - d[4, 1] d[2, 2] d[3, 3] d[1, 4]

         + d[4, 1] d[2, 2] d[1, 3] d[3, 4]

         - d[4, 1] d[3, 2] d[1, 3] d[2, 4]

         + d[4, 1] d[3, 2] d[2, 3] d[1, 4]

> ?Determinant
>