Hi Srinivasan Anand,
are you sure that the system is posted correctly. You seem to suggest that you are interested in a linear system, which yours is. But usually I would think of it as "vector = matrix times vector". Now the y is a vector, the c seems to be a matrix of coefficients and the x in your case is also a matrix, is it not? It that case the linear system would be:
Solve[{
y1 == c11*x1 + c12*x2 + c13*x3 + c14*x4,
y2 == c21*x1 + c22*x2 + c23*x3 + c24*x4,
y3 == c31*x1 + c32*x2 + c33*x3 + c34*x4,
y4 == c41*x1 + c42*x2 + c3*x3 + c44*x4}, {x1, x2, x3, x4}]
which gives a solution. In the case that you have posted things look differently. If I read your system correctly, all equations are independent. None of the variables occurs in more than one equation. Also, the
$x_{nj}$ will be underdetermined: one equation four variables. That means that you can get this relation:
Solve[yn == cn1*xn1 + cn2*xn2 + cn3*xn3 + cn4*xn4, {xn1, xn2, xn3, xn4}]
(*{{xn4 -> -((cn1 xn1)/cn4) - (cn2 xn2)/cn4 - (cn3 xn3)/cn4 + yn/cn4}}*)
The result is a relationship which allows you to determine one of the components of
$x$ based on three others and the coefficients of
$c$.
Perhaps I am missing something here...
Cheers,
Marco