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How to solve a matrix of Gaussian

solution by the Gauss algorithm

Gauss' method is one of the basic principles of solving a system of linear equations.

Its advantage is that it does not require the original quadratic matrix or pre-calculating its determinant.

You will need

  • The textbook on higher mathematics.



So you have a system of linear algebraic equations. This method has two main passages - forward and reverse.

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


Direct move: Record system in matrix vide.Sostavte extended matrix and bring it to echelon form usingelementary transformations of rows. It is worth recalling that the matrix has a stepped form, if the following two conditions: If any row of the matrix is ​​zero, then all subsequent lines are also nulevymi- support member of each subsequent line is to the right than in the rows predyduschey.Elementarnym conversion referred the following three steps type:
1) the locations of any permutation of two rows of the matrix.
2) replacement of any row sum of this string, with any other pre-multiplied by a number.
3) multiplying any row by a nonzerochislo.Opredelite rank of the augmented 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 augmented matrix, the system is not compatible and thus has no solutions. If the grades do not match, the system is consistent, and continue to search for solutions.

The matrix form of the system


Retraction:Declare basic unknowns are numbers that coincide with the numbers of basic columns of A (her step-type) and the other variables will be considered free. The number of free unknown is calculated by the formula k = n-r (A), where n-number of unknowns, r (A) is the rank of the matrix A.Dalee return to step matrix. Give it to the form of the Gauss. Recall that the speed matrix is ​​Gaussian if all its supporting elements equal to one, and over the support elements only zeros. Record system of algebraic equations, which corresponds to the matrix of the form of the Gauss, denoting the free unknowns as the C1, ..., express Ck.Na next step of the system obtained through free basic unknown.


Record your answer in vector-wise or form.

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