> 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]

> A := matrix(4,4,[a,b,0,1, b,a,0,0, 0,0,a,c, 1,0,c,a]);

                            [a    b    0    1]
                            [                ]
                            [b    a    0    0]
                       A := [                ]
                            [0    0    a    c]
                            [                ]
                            [1    0    c    a]

> mydet(A);

                    2  2    2  2    2    2  2    4
                   c  b  - c  a  - a  - a  b  + a

> ref(A);

           [a ,            b ,            0 ,            1]
           [                                              ]
           [               2                              ]
           [              b                               ]
           [0 ,        - ---- + a ,        0 ,       - b/a]
           [              a                               ]
           [                                              ]
           [0 ,            0 ,            a ,            c]
           [                                              ]
           [                2           2                 ]
           [               c           b                  ]
           [0 , 0 , 0 , - ---- - --------------- - 1/a + a]
           [               a        /    2     \          ]
           [                      2 |   b      |          ]
           [                     a  |- ---- + a|          ]
           [                        \   a      /          ]

> printlevel:=4:
> mydet(A);

         Elem step: add , - b/a,  times row , 1,  to row , 2


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


         Elem step: add , - 1/a,  times row , 1,  to row , 4


                [a        b         0        1    ]
                [                                 ]
                [         2                       ]
                [        b                        ]
                [0    - ---- + a    0      - b/a  ], a
                [        a                        ]
                [                                 ]
                [0        0         a        c    ]
                [                                 ]
                [0      - b/a       c    - 1/a + a]


           Elem step: add , 0,  times row , 2,  to row , 3


                            b
        Elem step: add , --------,  times row , 2,  to row , 4
                           2    2
                         -b  + a


        [a , b , 0 , 1]

        [        2                 ]
        [       b                  ]
        [0 , - ---- + a , 0 , - b/a]
        [       a                  ]

        [0 , 0 , a , c]

        [      /    2     \                                     ],
        [      |   b      |                                     ]
        [    b |- ---- + a|                     2               ]
        [      \   a      /                    b                ]
        [0 , -------------- - b/a , c , - ------------ - 1/a + a]
        [         2    2                     2    2             ]
        [       -b  + a                   (-b  + a ) a          ]

          2    2
        -b  + a


         Elem step: add , - c/a,  times row , 3,  to row , 4


        [a , b , 0 , 1]

        [        2                 ]
        [       b                  ]
        [0 , - ---- + a , 0 , - b/a]
        [       a                  ]

        [0 , 0 , a , c]

        [      /    2     \
        [      |   b      |
        [    b |- ---- + a|
        [      \   a      /
        [0 , -------------- - b/a , 0 ,
        [         2    2
        [       -b  + a

                                               2    2
                                       ], a (-b  + a )
                                       ]
            2          2               ]
           c          b                ]
        - ---- - ------------ - 1/a + a]
           a        2    2             ]
                 (-b  + a ) a          ]


        [a , b , 0 , 1]

        [        2                 ]
        [       b                  ]
        [0 , - ---- + a , 0 , - b/a]
        [       a                  ]

        [0 , 0 , a , c]

        [      /    2     \
        [      |   b      |
        [    b |- ---- + a|
        [      \   a      /
        [0 , -------------- - b/a , 0 ,
        [         2    2
        [       -b  + a

                                       ],
                                       ]
            2          2               ]
           c          b                ]
        - ---- - ------------ - 1/a + a]
           a        2    2             ]
                 (-b  + a ) a          ]

         2  2    2  2    2    2  2    4
        c  b  - c  a  - a  - a  b  + a


                    2  2    2  2    2    2  2    4
                   c  b  - c  a  - a  - a  b  + a

> B := matrix(E_I(4,2,4));

                            [1    0    0    0]
                            [                ]
                            [0    0    0    1]
                       B := [                ]
                            [0    0    1    0]
                            [                ]
                            [0    1    0    0]

> mydet(B);

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


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


           Elem step: add , 0,  times row , 1,  to row , 4


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


                   Elem step: exchange rows , 2, 4


           Elem step: add , 0,  times row , 2,  to row , 3


           Elem step: add , 0,  times row , 2,  to row , 4


                        [1    0    0    0]
                        [                ]
                        [0    1    0    0]
                        [                ], -1
                        [0    0    1    0]
                        [                ]
                        [0    0    0    1]


           Elem step: add , 0,  times row , 3,  to row , 4


                        [1    0    0    0]
                        [                ]
                        [0    1    0    0]
                        [                ], -1
                        [0    0    1    0]
                        [                ]
                        [0    0    0    1]


                        [1    0    0    0]
                        [                ]
                        [0    1    0    0]
                        [                ], -1
                        [0    0    1    0]
                        [                ]
                        [0    0    0    1]


                                  -1

> C := matrix(2,2,[a,b,c,d]);

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

> linalg[inverse](C);

                     [     d               b    ]
                     [ ---------     - ---------]
                     [ a d - b c       a d - b c]
                     [                          ]
                     [      c             a     ]
                     [- ---------     --------- ]
                     [  a d - b c     a d - b c ]

> b := vector(2,[alpha,beta]);

                          b := [alpha, beta]

> evalm(C^(-1) &* b);

           [ d alpha     b beta       c alpha     a beta  ]
           [--------- - ---------, - --------- + ---------]
           [a d - b c   a d - b c    a d - b c   a d - b c]

>