The Gauss method: examples of solutions and special cases

The Gaussian method, also called the step-by-step elimination method for unknown variables, is named after the distinguished German scientist K.F. Gauss, still in his lifetime, received the unofficial title of "king of mathematics." However, this method was known long before the birth of European civilization, back in the 1st century. BC. E. Ancient Chinese scientists used it in their writings.

The Gaussian method is a classical method for solving systems of linear algebraic equations (SLAE). It is ideal for quickly solving bounded matrices.

The method itself consists of two moves: direct and reverse. A straight run is a sequential casting of SLAU to a triangular form, that is, zeroing values located under the main diagonal. The reverse move implies the sequential finding of the values of the variables, expressing each variable through the previous one.

To learn how to apply the Gauss method in practice is simple, it is enough to know the elementary rules of multiplication, addition and subtraction of numbers.

In order to illustrate the algorithm for solving linear systems by this method, let us consider one example.

So, solve using the Gaussian method:

X + 2y + 4z = 3
2x + 6y + 11z = 6
4x-2y-2z = -6

We need to get rid of the variable x in the second and third lines. To do this, we add the first, multiplied by -2 and -4, respectively. We get:

X + 2y + 4z = 3
2y + 3z = 0
-10y-18z = -18

Now multiply the second line by 5 and add it to the third:

X + 2y + 4z = 3
2y + 3z = 0
-3z = -18

We brought our system to a triangular view. Now we are reversing. We start with the last line:
-3z = -18,
Z = 6.

Second line:
2y + 3z = 0
2y + 18 = 0
2y = -18,
Y = -9

First line:
X + 2y + 4z = 3
X-18 + 24 = 3
X = 18-24 + 3
Х = -3

Substituting the obtained values of the variables in the initial data, we are convinced of the correctness of the solution.

This example can be solved by many other substitutions, but the answer should be the same.

It happens that on the leading first line there are elements with too small values. It's not scary, but it's quite complicated. The solution to this problem is the method of Gauss with the choice of the main element by the column. Its essence consists in the following: in the first line the maximum element is found, the column in which it is located is interchanged with the 1-st column, that is, our maximal element becomes the first element of the main diagonal. Next is the standard calculation process. If necessary, the procedure for swapping columns can be repeated.

Another modified Gauss method is the Jordan-Gauss method.

It is used to solve square SLAU, when finding the inverse matrix and the rank of the matrix (the number of non-zero rows).

The essence of this method is that the original system transforms into a unit matrix by means of transformations with further finding the values of the variables.

Its algorithm is as follows:

1. The system of equations is reduced, as in the Gauss method, to a triangular form.

2. Each line is divided by a certain number so that the unit on the main diagonal is obtained.

3. The last line is multiplied by a certain number and subtracted from the penultimate one so that not on the main diagonal get 0.

4. The operation 3 is repeated successively for all rows until eventually a unit matrix is formed.