Solving systems of linear equations

Let A be a square matrix with n rows and n columns whose elements a _ij are known. Let b be a vector whose n components b _i are known. We search for the vector x, of components x ₁ , x ₂ , ..., x _n , which verifies the system of linear equations :

The solution to this problem is well known, and can be found in all linear algebra courses: x _i is equal to a ratio of determinants, with the determinant of the matrix A in the denominator and the same determinant in the numerator, in which...