Question

Назовите схему решения системы линейных уравнений по методу Гаусса.

Answer 1

Метод Гаусса, называемый также методом последовательного исключения неизвестных, состоит в том, что при помощи элементарных преобразований систему линейных уравнений приводят к такому виду, чтобы её матрица из коэффициентов оказалась трапециевидной или близкой к трапециевидной.

$a11x2 + a12x2 + a13x3 = b1 a22x2 + a23x3 = b2 a33x3 = b3$

Answer 2

Solving a System of Linear Equations using the Gauss Method

To solve a system of linear equations using the Gauss method, follow these steps:

1. Write the Augmented Matrix: - Write down the system of linear equations in the form of an augmented matrix, where the coefficients of the variables and the constants are arranged in a matrix.

2. Perform Row Operations: - Use row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. These operations include: - Interchanging two rows. - Multiplying a row by a nonzero number. - Adding a multiple of one row to another row.

3. Back Substitution: - Once the augmented matrix is in row-echelon form or reduced row-echelon form, use back substitution to solve for the variables.

4. Verify the Solution: - After obtaining the values of the variables, substitute them back into the original equations to verify the solution.

The Gauss method is a fundamental technique for solving systems of linear equations and is widely used in various fields, including mathematics, engineering, and computer science.