> restart;
> with(LinearAlgebra);

  [Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix,

        BidiagonalForm, BilinearForm, CharacteristicMatrix,

        CharacteristicPolynomial, Column, ColumnDimension,

        ColumnOperation, ColumnSpace, CompanionMatrix,

        ConditionNumber, ConstantMatrix, ConstantVector,

        CreatePermutation, CrossProduct, DeleteColumn, DeleteRow,

        Determinant, DiagonalMatrix, Dimension, Dimensions,

        DotProduct, Eigenvalues, Eigenvectors, Equal,

        ForwardSubstitute, FrobeniusForm, GaussianElimination,

        GenerateEquations, GenerateMatrix, GetResultDataType,

        GetResultShape, GivensRotationMatrix, GramSchmidt,

        HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm,

        HilbertMatrix, HouseholderMatrix, IdentityMatrix,

        IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar,

        IsUnitary, JordanBlockMatrix, JordanForm, LA_Main,

        LUDecomposition, LeastSquares, LinearSolve, Map, Map2,

        MatrixAdd, MatrixInverse, MatrixMatrixMultiply, MatrixNorm,

        MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial,

        Minor, Multiply, NoUserValue, Norm, Normalize, NullSpace,

        OuterProductMatrix, Permanent, Pivot, QRDecomposition,

        RandomMatrix, RandomVector, Rank, ReducedRowEchelonForm, Row,

        RowDimension, RowOperation, RowSpace, ScalarMatrix,

        ScalarMultiply, ScalarVector, SchurForm, SingularValues,

        SmithForm, SubMatrix, SubVector, SumBasis, SylvesterMatrix,

        ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector,

        VandermondeMatrix, VectorAdd, VectorAngle,

        VectorMatrixMultiply, VectorNorm, VectorScalarMultiply,

        ZeroMatrix, ZeroVector, Zip]

> A := <<a,b>|<c,d>>;

                                 [a    c]
                            A := [      ]
                                 [b    d]

> B := MatrixInverse(A);

                       [     d               c    ]
                       [ ---------     - ---------]
                       [ a d - c b       a d - c b]
                  B := [                          ]
                       [      b             a     ]
                       [- ---------     --------- ]
                       [  a d - c b     a d - c b ]

> map(simplify,A . B);

                               [1    0]
                               [      ]
                               [0    1]

> map(simplify,B . A);

                               [1    0]
                               [      ]
                               [0    1]

> v := Vector([alpha, beta]);

                                  [alpha]
                             v := [     ]
                                  [beta ]

> x := B . v;

                         [  d alpha     c beta   ]
                         [ --------- - --------- ]
                         [ a d - c b   a d - c b ]
                    x := [                       ]
                         [   b alpha     a beta  ]
                         [- --------- + ---------]
                         [  a d - c b   a d - c b]

> map(simplify, A . x);

                               [alpha]
                               [     ]
                               [beta ]

> 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.ncsu.edu/dist/maple70/lib"

> with(RefPkg);

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

> ?E_I
> E1 := E_I(5,2,4);

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

> A1 := <<a,b,c,d,e>|<1,2,3,4,5>>;

                                  [a    1]
                                  [      ]
                                  [b    2]
                                  [      ]
                            A1 := [c    3]
                                  [      ]
                                  [d    4]
                                  [      ]
                                  [e    5]

> E1 . A1;

                               [a    1]
                               [      ]
                               [d    4]
                               [      ]
                               [c    3]
                               [      ]
                               [b    2]
                               [      ]
                               [e    5]

> A . E_I(2,1,2);

                               [c    a]
                               [      ]
                               [d    b]

> E2:=E_II(5,4,gamma);

                        [1    0    0      0      0]
                        [                         ]
                        [0    1    0      0      0]
                        [                         ]
                  E2 := [0    0    1      0      0]
                        [                         ]
                        [0    0    0    gamma    0]
                        [                         ]
                        [0    0    0      0      1]

> E2 . A1;

                         [   a          1   ]
                         [                  ]
                         [   b          2   ]
                         [                  ]
                         [   c          3   ]
                         [                  ]
                         [gamma d    4 gamma]
                         [                  ]
                         [   e          5   ]

> Transpose(A1) . E2;

                    [a    b    c    gamma d    e]
                    [                           ]
                    [1    2    3    4 gamma    5]

> E3 := E_III(5,2,4,delta);

                        [1      0      0    0    0]
                        [                         ]
                        [0      1      0    0    0]
                        [                         ]
                  E3 := [0      0      1    0    0]
                        [                         ]
                        [0    delta    0    1    0]
                        [                         ]
                        [0      0      0    0    1]

> E3 . A1;

                     [     a              1     ]
                     [                          ]
                     [     b              2     ]
                     [                          ]
                     [     c              3     ]
                     [                          ]
                     [delta b + d    4 + 2 delta]
                     [                          ]
                     [     e              5     ]

> A2 := Transpose(A1);

                          [a    b    c    d    e]
                    A2 := [                     ]
                          [1    2    3    4    5]

> A2 . E3;

                  [a    b + d delta    c    d    e]
                  [                               ]
                  [1    2 + 4 delta    3    4    5]

> E1;

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

> MatrixInverse(E1);

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

> MatrixInverse(E2);

                     [1    0    0      0      0]
                     [                         ]
                     [0    1    0      0      0]
                     [                         ]
                     [0    0    1      0      0]
                     [                         ]
                     [                 1       ]
                     [0    0    0    -----    0]
                     [               gamma     ]
                     [                         ]
                     [0    0    0      0      1]

> MatrixInverse(E3);

                     [1      0       0    0    0]
                     [                          ]
                     [0      1       0    0    0]
                     [                          ]
                     [0      0       1    0    0]
                     [                          ]
                     [0    -delta    0    1    0]
                     [                          ]
                     [0      0       0    0    1]

> A3 := Matrix([[0,2,-5],[0,-1,3],[1,0,0]]);

                              [0     2    -5]
                              [             ]
                        A3 := [0    -1     3]
                              [             ]
                              [1     0     0]

> MatrixInverse(A3);

                            [0    0    1]
                            [           ]
                            [3    5    0]
                            [           ]
                            [1    2    0]

>