Solve a system of 4 algebraic equations?

[tnaime (tarek naime)]

I’m trying to solve a system of 4 algebraic equation, but I don’t know why I have this error

    Solve[{
      0 == (a12 *x2 + a13 *x3 + a14* x4) - x1*(a21 + a31 + a41) , 
      0 == (a21 *x1 + a23* x3 + a24* x4) - x2*(a12 + a32 + a42), 
      0 == (a32* x2 + a31* x1 + a34 *x4) - x3*(a23 + a13 + a43), 
      0 == (a42* x2 + a43 *x3 + a41 *x1) - x4*(a24 + a34 + a14)}, {x1, x2,
       x3, x4}]

Equations may not give solutions for all "solve" variables. >>

this error comes when I have more equations than variables, but this is not the case.

[wufei1978 (Frederick Wu)]

equations = {0 == (a12 x2 + a13 x3 + a14 x4) - x1 (a21 + a31 + a41),
0 == (a21 x1 + a23 x3 + a24 x4) - x2 (a12 + a32 + a42),
0 == (a32 x2 + a31 x1 + a34 x4) - x3 (a23 + a13 + a43),
0 == (a42 x2 + a43 x3 + a41 x1) - x4 (a24 + a34 + a14)};

Your equations looks like 4 equations, but actually only 3 independent equations.
It can be coverted to A*X=0 in linear algebra. A matrix coefficient can be taken by below code.

Last@Normal@CoefficientArrays[equations, {x1, x2, x3, x4}]
MatrixRank[%]

The matrix rank should be 4 as a completely solveable linear equations, but your equations rank is only 3.
Therefore, Mathematica Solve funtion gives that errors message.