The classical Gaussian method  is the most efficient and universal for solving a system of linear algebraic equations A.X=B. For particular cases when A is a non-singular square matrix the solution can be written as X=A^(-1).B. This is equivalent to the Cramer's rule https://en.wikipedia.org/wiki/Cramer%27s_rule

Cramers Rule

To solve  Ax = b  explicitly by Cramer's rule, it costs n! multiplications.

I'll join here a system of equations i am trying to solve, the uknowns are the Cij (i,j=1,4), the coefficients Dijk (i,j=1,4, k=1,12) are functions (mainly function of the Fourier variable Lamda) and F is another function, i did it by ellimination and then tried on Maple, it doesn't give solution, by hand i got verry complicated solutions, my aim is to reduce the numbre of muliplied terms so i can simplify the solutions. Thanks 

Dear Brick chaouche Amine Amine,

The system of linear equations above. After transform to an augmented matrix (see attachment) then reduce the matrix to "Reduced row echelon form" you may get the solutions.

Please see the following attachment.

Regards,

Wiwat Wanicharpichat

Thanks so much Mr Wiwat Wanicharpichat 

Dear Brick chaouche Amine Amine,

You are welcome, glad to be of service.

Regards,

Wiwat Wanicharpichat

Dear Brick

The Krakowian calculus which involves a column-by-column multiplication of matrices (introduced by Tadeusz Banachiewicz) provides to very fast and cheap solution of linear system of equations.

Greetings, Tadeusz

Thanks Mr Tadeusz Ostrowski, in fact i have to solve a system of equations in which the coefficients are not constants and are function of the Fourier variable Lampda and some coefficients involve a complex parts, i have to solve 10 equations to find 10 unknowns, i tried to handle it using the classical methode of ellimination and it yields a very complicated terms and very large expressions that i have to simplify using Maple or Mathematica, i am working on it and i can send a document about that if you would like to help me with that or to see the form of the expressions i have. thank you