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How to solve the Gaussian matrix


Gauss algorithm

The Gaussian method is one of the basic principles of solving a system of linear equations.

Its advantage lies in the fact that it does not require the quadraticity of the original matrix or the preliminary calculation of its determinant.

You will need

  • Textbook on higher mathematics.



So you have a system of linear algebraic equations. This method consists of two main moves - direct and reverse.

How to solve & lt-strong & gt-matrix & lt- / strong & gt- by & lt-b & gt-Gauss & lt- / b & gt-


Straight move: Write the system in the matrix view. Matrix And bring it to a stepped view withElementary string transformations. It should be recalled that the matrix has a step-like form if the following two conditions are fulfilled: If any row of the matrix is ​​zero, then all subsequent lines are also zero. The supporting element of each subsequent line is to the right of the previous one. The elementary transformation of the rows is the actions of the following three Types:
1) permutation of any two rows of the matrix.
2) the replacement of any line with the sum of this line from any other, previously multiplied by a certain number.
3) multiplying any string by a non-zeroDetermine the rank of the expanded matrix and draw a conclusion about the compatibility of the system. If the rank of the matrix A does not coincide with the rank of the extended matrix, then the system is not compatible and accordingly has no solution. If the ranks do not match, then the system is compatible, and continue to search for solutions.

The matrix form of the system


Reverse:Declare the basic unknowns those whose numbers coincide with the numbers of the base columns of the matrix A (its stepped form), and the remaining variables will be considered free. The number of free unknowns is calculated by the formula k = n-r (A), where n is the number of unknowns, r (A) is the matrix of A. Next, return to the step matrix. Bring her to the sight of Gauss. Recall that the step matrix has the form of a Gaussian if all its support elements are equal to one, and above the support elements are some zeros. Write down a system of algebraic equations that corresponds to a matrix of the Gauss type, denoting free unknowns as C1, ..., Ck. In the next step, express the basic unknowns from the obtained system in terms of free ones.


Record the answer in vector or coordinate form.

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