Solving systems of linear equations on Gauss's method

Solving a system of n equations with n unknowns by Cramer's rule, we must compute n +1 determinants of order n. This is a hard work.

Moreover, the method of Cramer cannot be used in cases where the principal determinant equals zero or the number of equations does not agree with that of unknowns. In such cases, Gauss' method of successive elimination of unknowns extended by applying matrices is used.

Consider the Gauss method in the case where the number of equations coincides with that of unknowns (3).

Suppose that а ₁₁¹0; let us divide the first equation by this coefficient:

. (*)

Multiplying the resulting equation by – а ₂₁ and adding it to the second equation of system (3), we obtain

.

Similarly, multiplying equation (*) by – а _n1 and adding it to the last equation of system (3), we obtain

.

At the end, we obtain the new system of equations with n – 1 unknowns:

(4)

System (4) is obtained from system (3) by applying linear transformations of equations; hence this system is equivalent to (3), i.e., any solution of system (4) is a solution of the initial system of equations.

To get rid of х ₂ in the third, the forth, …, n th-equation, we multiply the second equation of system (4) by and, multiplying this equation by the negative coefficients of х ₂ and summing them, obtain

Performing this procedure n times, we reduce the system of equations to the diagonal form

We determine х_n from the last equation, substitute it in the preceding equation and obtain x_{n -1}, and so on; going up, we determine х ₁ from the first equation. This is the classical Gauss method.

Поиск по сайту: