Solving symbolic linear equations with maple

Question

How can I solve linear equations of the following type in Maple?

$$\begin{pmatrix} 1 & 1 & 1 & 1\\ b-c & c-b & a-b &0 \\ b-d & d-a & 0 &a-b \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{pmatrix} =\begin{pmatrix} 0\\ 0 \\ 0 \end{pmatrix}$$

The problem is that I want to use $a,b,c,d$ as symbolic variables. It seems, Maple only can solve such an equation when $a,b,c,d$ are fixed numbers.

Answer 1

I don't see Maple having any problem with generating a symbolic solution to this under-determined system, using either the LinearSolve or the solve commands.

restart:

M := Matrix( [[1,   1,   1,   1],
              [b-c, c-b, a-b, 0],
              [b-d, d-a, 0, a-b]] );

                       [  1      1      1      1  ]
                       [                          ]
                  M := [b - c  c - b  a - b    0  ]
                       [                          ]
                       [b - d  d - a    0    a - b]

B := Vector([0, 0, 0]);

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

Unsurprisingly we get a solution with one free parameter.

X := LinearAlgebra:-LinearSolve(M, B);

                        [   _t[3] (2 a - c - d) ]
                        [ - ------------------- ]
                        [        3 (b - c)      ]
                        [                       ]
                        [_t[3] (a - 3 b + c + d)]
                        [-----------------------]
                   X := [       3 (b - c)       ]
                        [                       ]
                        [         _t[3]         ]
                        [                       ]
                        [  (a + c - 2 d) _t[3]  ]
                        [  -------------------  ]
                        [       3 (b - c)       ]

map( simplify, M . X - B );

                                 [0]
                                 [ ]
                                 [0]
                                 [ ]
                                 [0]

The above is all in Matrix-Vector form. Equation form can also be produced straightforwardly.

Xsymb := Vector([x1,x2,x3,x4]);

                                     [x1]
                                     [  ]
                                     [x2]
                            Xsymb := [  ]
                                     [x3]
                                     [  ]
                                     [x4]

ans := Equate( Xsymb, X );

           [       _t[3] (2 a - c - d)       _t[3] (a - 3 b + c + d)              
    ans := [x1 = - -------------------, x2 = -----------------------, x3 = _t[3], 
           [            3 (b - c)                   3 (b - c)                     

                   (a + c - 2 d) _t[3]]
              x4 = -------------------]
                        3 (b - c)     ]

Or, we could reformulate the question in terms of explicit equations, and pass those to the solve command.

LinearAlgebra:-GenerateEquations( Matrix([M,B]), convert(Xsymb,list) );

        [x1 + x2 + x3 + x4 = 0, a x3 + b x1 - b x2 - b x3 - c x1 + c x2 = 0, 

          -a x2 + a x4 + b x1 - b x4 - d x1 + d x2 = 0]

solve( %, [x1, x2, x4] );

        [[       x3 (2 a - c - d)       x3 (a - 3 b + c + d)       x3 (a + c - 2 d)]]
        [[x1 = - ----------------, x2 = --------------------, x4 = ----------------]]
        [[          3 (b - c)                3 (b - c)                3 (b - c)    ]]