Gauss's method is one of the basic principles of the solution of a system of the linear equations. Its advantage is that it does not demand a kvadratichnost of an initial matrix or preliminary calculation of its determinant.
It is required to you
- The textbook on the higher mathematics.
Instruction
1. So you have a system of the linear algebraic equations. This method consists of two main courses - direct and the return.
2. Forward stroke: Write down a system in a matrix look. Make an expanded matrix and lead it to a step view with the help of elementary transformations of lines. It is worth reminding that the matrix has a step appearance if the following two conditions are satisfied: If any line of a matrix zero, then all next lines are zero too; The Basic element of everyone the subsequent lines is more to the right, than in previous. Elementary transformation of lines call actions of the following three types: 1) shift by places of any two lines of a matrix.2) replacement of any line with the sum of this line with any other, previously increased by some number.3) multiplication of any line by number, other than zero. Define a rank of an expanded matrix and draw a conclusion on compatibility of a system. If the rank a matrix But does not coincide with a rank of an expanded matrix, then the system not in common and respectively has no decision. If ranks do not coincide, then a system in common, and continue finding solutions.
3. Back run: Declare basic unknown those which numbers will coincide with numbers of basic columns of a matrix And (its step look), and you will consider other variables free. We find number of free unknown on a formula k=n-r (A) where n-number of unknown, r(A) - a rank a matrix A. Daley return to a step matrix. Lead it to Gauss's type. Let's remind that the step matrix has Gauss's appearance if its all basic elements are equal to unit, and over basic elements some zero. Write down the system of the algebraic equations which corresponds to a matrix of a type of Gauss, having designated free unknown as C1..., Ck. On the following step express from the received system basic unknown through free.
4. Write down the answer in a vector or pokoordinatny look.