be a system of n simultaneous first-degree equations in n unknowns, x₁, x₂, ..., x_n. The determinant D of (4) is the determinant of the system for (12).

Let D₁, D₂, ..., D_n be the determinants that result when the column of b's in (12) is substituted for the first, second, ..., nth column, respectively, of the D of (4). The general Cramer's Rule states that if then the system (12) has the unique solution

and that if D = 0, then the system either has no solution or an infinite number of solutions. We do not give the proof of this rule for general n.

When the number of equations in (12) is large, it becomes very difficult to evaluate the n + 1 determinants in (13) by the methods discussed in this book. More advanced texts describe variations of the elimination techniques that are practical for the numerical approximation of determinants of large order or in the solution of systems (12) with n large. (For example, see the description of Crout's Method in the appendix of F. B. Hildebrand's Methods of Applied Mathematics, Prentice-Hall, 1952.)
 
 

Tuesday, June 23, 1998