Carl Friedrich Gauss - German mathematician, founder of the SLAE method of the same name
Carl Friedrich Gauss was a famous great mathematician and he was once recognized as the “king of mathematics”. Although the name "Gauss method" is generally accepted, Gauss is not its author: the Gauss method was known long before him. Its first description is in the Chinese treatise Mathematics in Nine Books, which was compiled between the 2nd century BC. BC e. and I century. n. e. and is a compilation of earlier works written around the 10th century. BC e.
– consistent exclusion of unknowns. This method is used to solve quadratic systems of linear algebraic equations. Although equations are easily solved using the Gauss method, students often cannot find the correct solution because they get confused in signs (pluses and minuses). Therefore, when solving SLAE, it is necessary to be extremely careful and only then can even the most complex equation be easily, quickly and correctly solved.
Systems of linear algebraic equations have several advantages: the equation is not necessarily consistent in advance; it is possible to solve such systems of equations in which the number of equations does not coincide with the number of unknown variables or the determinant of the main matrix is equal to zero; it is possible to use the Gaussian method to lead to a result with a relatively small number of computational operations.
As already mentioned, the Gauss method causes some difficulties for students. However, if you learn the methodology and algorithm of the solution, you will immediately understand the intricacies of the solution.
First, let's systematize knowledge about systems of linear equations.
Note!
Depending on its elements, an SLAE can have:
- One solution;
- many solutions;
- have no solutions at all.
In the first two cases, the SLAE is called compatible, and in the third case, it is called incompatible. If the system has one solution, it is called definite, and if there are more than one solution, then the system is called indefinite.
Gauss method - theorem, examples of solutions updated: November 22, 2019 by: Scientific Articles.Ru
Carl Friedrich Gauss, the greatest mathematician, hesitated for a long time, choosing between philosophy and mathematics. Perhaps it was precisely this mindset that allowed him to make such a noticeable “legacy” in world science. In particular, by creating the "Gauss Method" ...
For almost 4 years, articles on this site dealt with school education, mainly from the point of view of philosophy, the principles of (mis)understanding introduced into the minds of children. The time is coming for more specifics, examples and methods... I believe that this is exactly the approach to the familiar, confusing and important areas of life gives better results.
We people are designed in such a way that no matter how much we talk about abstract thinking, But understanding Always happens through examples. If there are no examples, then it is impossible to grasp the principles... Just as it is impossible to get to the top of a mountain except by walking the entire slope from the foot.
Same with school: for now living stories It is not enough that we instinctively continue to regard it as a place where children are taught to understand.
For example, teaching the Gaussian method...
Gauss method in 5th grade school
I’ll make a reservation right away: the Gauss method has a much wider application, for example, when solving systems of linear equations. What we will talk about takes place in 5th grade. This started, having understood which, it is much easier to understand the more “advanced options”. In this article we are talking about Gauss's method (method) for finding the sum of a series
Here is an example that my youngest son, who attends 5th grade at a Moscow gymnasium, brought from school.
School demonstration of the Gauss method
Math teacher using interactive whiteboard ( modern methods training) showed the children a presentation of the history of the “creation of the method” by little Gauss.
The school teacher whipped little Karl (an outdated method, not used in schools these days) because he
instead of sequentially adding numbers from 1 to 100, find their sum noticed that pairs of numbers equally spaced from the edges of an arithmetic progression add up to the same number. for example, 100 and 1, 99 and 2. Having counted the number of such pairs, little Gauss almost instantly solved the problem proposed by the teacher. For which he was executed in front of an astonished public. So that others would be discouraged from thinking.
What did little Gauss do? developed number sense? Noticed some feature number series with a constant step (arithmetic progression). AND exactly this later made him a great scientist, those who know how to notice, having feeling, instinct of understanding.
This is why mathematics is valuable, developing ability to see general in particular - abstract thinking. Therefore, most parents and employers instinctively consider mathematics an important discipline ...
“Then you need to learn mathematics, because it puts your mind in order.
M.V.Lomonosov".
However, the followers of those who flogged future geniuses with rods turned the Method into something the opposite. As my supervisor said 35 years ago: “The question has been learned.” Or as my youngest son said yesterday about Gauss’s method: “Maybe it’s not worth making a big science out of this, huh?”
The consequences of the creativity of the “scientists” are visible in the level of current school mathematics, the level of its teaching and the understanding of the “Queen of Sciences” by the majority.
However, let's continue...
Methods for explaining the Gauss method in 5th grade school
A mathematics teacher at a Moscow gymnasium, explaining the Gauss method according to Vilenkin, complicated the task.
What if the difference (step) of an arithmetic progression is not one, but another number? For example, 20.
The problem he gave to the fifth graders:
20+40+60+80+ ... +460+480+500
Before getting acquainted with the gymnasium method, let’s take a look at the Internet: how do school teachers and math tutors do it?..
Gaussian method: explanation No. 1
A well-known tutor on his YOUTUBE channel gives the following reasoning:
"Let's write the numbers from 1 to 100 as follows:
first a series of numbers from 1 to 50, and strictly below it another series of numbers from 50 to 100, but in the reverse order"
1, 2, 3, ... 48, 49, 50
100, 99, 98 ... 53, 52, 51
"Please note: the sum of each pair of numbers from the top and bottom rows is the same and equals 101! Let's count the number of pairs, it is 50 and multiply the sum of one pair by the number of pairs! Voila: The answer is ready!"
“If you couldn’t understand, don’t be upset!” the teacher repeated three times during the explanation. "You will take this method in 9th grade!"
Gaussian method: explanation No. 2
Another tutor, less well-known (judging by the number of views), takes a more scientific approach, offering a solution algorithm of 5 points that must be completed sequentially.
For the uninitiated, 5 is one of the Fibonacci numbers traditionally considered magical. A 5 step method is always more scientific than a 6 step method, for example. ...And this is hardly an accident, most likely, the Author is a hidden adherent of the Fibonacci theory
Given an arithmetic progression: 4, 10, 16 ... 244, 250, 256 .
Algorithm for finding the sum of numbers in a series using the Gauss method:
4, 10, 16 ... 244, 250, 256
256, 250, 244 ... 16, 10, 4
At the same time, you need to remember plus one rule : we must add one to the resulting quotient: otherwise we will get a result that is less by one than the true number of pairs: 42 + 1 = 43.
This is the required sum of the arithmetic progression from 4 to 256 with a difference of 6!
Gauss method: explanation in 5th grade at a Moscow gymnasium
Here's how to solve the problem of finding the sum of a series:
20+40+60+ ... +460+480+500
in the 5th grade of a Moscow gymnasium, Vilenkin’s textbook (according to my son).
After showing the presentation, the math teacher showed a couple of examples using the Gaussian method and gave the class a task of finding the sum of the numbers in a series in increments of 20.
This required the following:
As you can see, this is a more compact and effective technique: the number 3 is also a member of the Fibonacci sequence
My comments on the school version of the Gauss method
The great mathematician would definitely have chosen philosophy if he had foreseen what his “method” would be turned into by his followers German teacher, who flogged Karl with rods. He would have seen the symbolism, the dialectical spiral and the undying stupidity of the “teachers”, trying to measure the harmony of living mathematical thought with the algebra of misunderstanding ....
By the way: did you know. that our education system is rooted in the German school of the 18th and 19th centuries?
But Gauss chose mathematics.
What is the essence of his method?
IN simplification. IN observing and grasping simple patterns of numbers. IN turning dry school arithmetic into interesting and exciting activity , activating in the brain the desire to continue, rather than blocking high-cost mental activity.
Is it possible to use one of the given “modifications of Gauss’s method” to calculate the sum of the numbers of an arithmetic progression almost instantly? According to the “algorithms”, little Karl would be guaranteed to avoid spanking, develop an aversion to mathematics and suppress his creative impulses in the bud.
Why did the tutor so persistently advise fifth-graders “not to be afraid of misunderstanding” of the method, convincing them that they would solve “such” problems as early as 9th grade? Psychologically illiterate action. It was a good move to note: "See? You already in 5th grade you can solve problems that you will complete only in 4 years! What a great fellow you are!”
To use the Gaussian method, a level of class 3 is sufficient, when normal children already know how to add, multiply and divide 2-3 digit numbers. Problems arise due to the inability of adult teachers who are “out of touch” to explain the simplest things in normal human language, not to mention mathematical... They are unable to get people interested in mathematics and completely discourage even those who are “capable.”
Or, as my son commented: “making a big science out of it.”
Gauss method, my explanations
My wife and I explained this “method” to our child, it seems, even before school...
Simplicity instead of complexity or a game of questions and answers
"Look, here are the numbers from 1 to 100. What do you see?"
The point is not what exactly the child sees. The trick is to get him to look.
"How can you put them together?" The son realized that such questions are not asked “just like that” and you need to look at the question “somehow differently, differently than he usually does”
It doesn't matter if the child sees the solution right away, it's unlikely. It is important that he stopped being afraid to look, or as I say: “moved the task”. This is the beginning of the journey to understanding
“Which is easier: adding, for example, 5 and 6 or 5 and 95?” A leading question... But any training comes down to “guiding” a person to the “answer” - in any way acceptable to him.
At this stage, guesses may already arise about how to “save” on calculations.
All we did was hint: the “frontal, linear” method of counting is not the only possible one. If a child understands this, then later he will come up with many more such methods, because it's interesting!!! And he will definitely avoid “misunderstanding” mathematics and will not feel disgusted with it. He got the win!
If child discovered that adding pairs of numbers that add up to a hundred is a piece of cake, then "arithmetic progression with difference 1"- a rather dreary and uninteresting thing for a child - suddenly found life for him . Order emerged from chaos, and this always causes enthusiasm: that's how we are made!
A question to answer: why, after the insight a child has received, should he again be forced into the framework of dry algorithms, which are also functionally useless in this case?!
Why force stupid rewrites? sequence numbers in a notebook: so that even the capable do not have a single chance of understanding? Statistically, of course, but mass education is geared towards “statistics”...
Where did zero go?
And yet, adding numbers that add up to 100 is much more acceptable to the mind than those that add up to 101...
The "Gauss School Method" requires exactly this: mindlessly fold pairs of numbers equidistant from the center of the progression, Despite everything.
What if you look?
Still, zero is the greatest invention of mankind, which is more than 2,000 years old. And math teachers continue to ignore him.
It's much easier to convert a series of numbers starting at 1 into a series starting at 0. The sum won't change, will it? You need to stop "thinking in textbooks" and start looking ... And to see that pairs with sum 101 can be completely replaced by pairs with sum 100!
0 + 100, 1 + 99, 2 + 98 ... 49 + 51
How to abolish the "rule plus 1"?
To be honest, I first heard about such a rule from that YouTube tutor ...
What do I still do when I need to determine the number of members of a series?
Looking at the sequence:
1, 2, 3, .. 8, 9, 10
and when completely tired, then on a simpler row:
1, 2, 3, 4, 5
and I figure: if you subtract one from 5, you get 4, but I'm quite clear I see 5 numbers! Therefore, you need to add one! Number sense developed in primary school, suggests: even if there are a whole Google of members of the series (10 to the hundredth power), the pattern will remain the same.
What the hell are the rules?..
So that in a couple of - three years to fill all the space between the forehead and the back of the head and stop thinking? How to earn your bread and butter? After all, we are moving in even ranks into the era of the digital economy!
More about Gauss’s school method: “why make science out of this?..”
It was not for nothing that I posted a screenshot from my son’s notebook...
"What happened in class?"
“Well, I counted right away, raised my hand, but she didn’t ask. Therefore, while the others were counting, I began to do homework in Russian so as not to waste time. Then, when the others finished writing (???), she called me to the board. I said the answer."
“That’s right, show me how you solved it,” said the teacher. I showed it. She said: “Wrong, you need to count as I showed!”
“It’s good that she didn’t give a bad grade. And she made me write in their notebook “the course of the solution” in their own way. Why make a big science out of this?..”
The main crime of a math teacher
Hardly after that incident Carl Gauss experienced a high sense of respect for the school teacher of mathematics. But if he knew how followers of that teacher pervert the essence of the method...he would roar with indignation and through the World Organization intellectual property WIPO has achieved a ban on the use of his honest name in school textbooks! ..
In what the main mistake of the school approach? Or, as I put it, a crime of school mathematics teachers against children?
Algorithm of misunderstanding
What do school methodologists do, the vast majority of whom do not know how to think?
They create methods and algorithms (see). This a defensive reaction that protects teachers from criticism (“Everything is done according to...”) and children from understanding. And thus - from the desire to criticize teachers!(The second derivative of bureaucratic “wisdom”, a scientific approach to the problem). A person who does not grasp the meaning will rather blame his own misunderstanding, rather than the stupidity of the school system.
This is what happens: parents blame their children, and teachers... do the same for children who “don’t understand mathematics!”
Are you smart?
What did little Karl do?
A completely unconventional approach to a formulaic task. This is the essence of His approach. This the main thing that should be taught in school is to think not with textbooks, but with your head. Of course, there is also an instrumental component that can be used... in search of simpler and effective methods accounts.
Gauss method according to Vilenkin
In school they teach that Gauss's method is to
What, if the number of elements of the series is odd, as in the problem that was assigned to my son?..
The "catch" is that in this case you should find an “extra” number in the series and add it to the sum of the pairs. In our example this number is 260.
How to detect? Copying all pairs of numbers into a notebook!(This is why the teacher made the kids do this stupid job of trying to teach "creativity" using the Gaussian method... And this is why such a "method" is practically inapplicable to large data series, AND this is why it is not the Gaussian method.)
A little creativity in the school routine...
The son acted differently.
(20 + 500, 40 + 480 ...).
0+500, 20+480, 40+460 ...
Not difficult, right?
But in practice it becomes even easier, which allows you to carve out 2-3 minutes for remote sensing in Russian, while the rest are "counting". In addition, it retains the number of steps of the methodology: 5, which does not allow criticizing the approach for being unscientific.
Obviously this approach is simpler, faster and more versatile, in the style of the Method. But ... the teacher not only did not praise, but also made me rewrite " in the right way"(See screenshot). That is, she made a desperate attempt to stifle the creative impulse and the ability to understand mathematics in the bud! Apparently, in order to later get hired as a tutor ... She attacked the wrong one ...
Everything that I have described so long and tediously can be explained to a normal child in a maximum of half an hour. Along with examples.
And in such a way that he will never forget it.
And it will be step towards understanding...not just mathematicians.
Admit it: how many times in your life have you added using the Gaussian method? And I never did!
But instinct of understanding, which develops (or is extinguished) in the process of learning mathematical methods at school... Oh!.. This is truly an irreplaceable thing!
Especially in the age of universal digitalization, which we have quietly entered under the strict leadership of the Party and the Government.
A few words in defense of teachers...
It is unfair and wrong to place all responsibility for this style of teaching solely on school teachers. The system is in effect.
Some teachers understand the absurdity of what is happening, but what to do? The Law on Education, Federal State Educational Standards, methods, lesson plans... Everything must be done “in accordance and on the basis” and everything must be documented. Step aside - stood in line to be fired. Let’s not be hypocrites: the salaries of Moscow teachers are very good... If they fire you, where to go?..
Therefore this site not about education. He's about individual education, only possible way get out of the crowd generation Z ...
Let the system be given, ∆≠0. (1)Gauss method is a method of sequentially eliminating unknowns.
The essence of the Gauss method is to transform (1) to a system with a triangular matrix, from which the values of all unknowns are then obtained sequentially (in reverse). Let's consider one of the computational schemes. This circuit is called a single division circuit. So let's look at this diagram. Let a 11 ≠0 (leading element) divide the first equation by a 11. We get
x 1 +a (1) 12 x 2 +...+a (1) 1n x n =b (1) 1 (2)
Using equation (2), it is easy to eliminate the unknowns x 1 from the remaining equations of the system (to do this, it is enough to subtract equation (2) from each equation, previously multiplied by the corresponding coefficient for x 1), that is, in the first step we obtain
.
In other words, at step 1, each element of subsequent rows, starting from the second, is equal to the difference between the original element and the product of its “projection” onto the first column and the first (transformed) row.
Following this, leaving the first equation alone, we perform a similar transformation over the remaining equations of the system obtained in the first step: we select from among them the equation with the leading element and, with its help, exclude x 2 from the remaining equations (step 2).
After n steps, instead of (1), we obtain an equivalent system 
(3)
Thus, at the first stage we obtain a triangular system (3). This stage is called forward stroke.
At the second stage (reverse), we find sequentially from (3) the values x n, x n -1, ..., x 1.
Let us denote the resulting solution as x 0 . Then the difference ε=b-A x 0 is called residual.
If ε=0, then the found solution x 0 is correct.
Calculations using the Gaussian method are performed in two stages:
- The first stage is called the forward method. At the first stage, the original system is converted to a triangular form.
- The second stage is called the reverse stroke. At the second stage, a triangular system equivalent to the original one is solved.
At each step, it was assumed that the leading element is different from zero. If this is not the case, then any other element can be used as a leader, as if rearranging the equations of the system.
Purpose of the Gauss method
The Gauss method is intended for solving systems of linear equations. Refers to direct methods of solution.Types of Gaussian method
- Classical Gauss method;
- Modifications of the Gauss method. One of the modifications of the Gaussian method is the circuit with the choice of the main element. A feature of the Gauss method with the choice of the main element is such a permutation of the equations so that at the k-th step the leading element is the largest element in the k-th column.
- Jordano-Gauss method;
Let's illustrate the difference Jordano-Gauss method from the Gaussian method with examples.
Example of a solution using the Gaussian method
Let's solve the system: 
Let's multiply the 2nd line by (2). Add the 3rd line to the 2nd 
From the 1st line we express x 3:
From the 2nd line we express x 2:
From the 3rd line we express x 1: 
An example of a solution using the Jordano-Gauss method
Let us solve the same SLAE using the Jordano-Gauss method. 
We will sequentially select the resolving element RE, which lies on the main diagonal of the matrix.
The resolution element is equal to (1).
NE = SE - (A*B)/RE
RE - resolving element (1), A and B - matrix elements forming a rectangle with elements STE and RE.
Let's present the calculation of each element in the form of a table:
| x 1 | x 2 | x 3 | B |
| 1 / 1 = 1 | 2 / 1 = 2 | -2 / 1 = -2 | 1 / 1 = 1 |
 |  |  |  |
The resolving element is equal to (3).
In place of the resolving element we get 1, and in the column itself we write zeros.
All other elements of the matrix, including elements of column B, are determined by the rectangle rule.
To do this, we select four numbers that are located at the vertices of the rectangle and always include the resolving element RE.
| x 1 | x 2 | x 3 | B |
 |  |
||
| 0 / 3 = 0 | 3 / 3 = 1 | 1 / 3 = 0.33 | 4 / 3 = 1.33 |
 |
The resolution element is (-4).
In place of the resolving element we get 1, and in the column itself we write zeros.
All other elements of the matrix, including elements of column B, are determined by the rectangle rule.
To do this, we select four numbers that are located at the vertices of the rectangle and always include the resolving element RE.
Let's present the calculation of each element in the form of a table:
| x 1 | x 2 | x 3 | B |
 |  |  |  |
 |  |
||
| 0 / -4 = 0 | 0 / -4 = 0 | -4 / -4 = 1 | -4 / -4 = 1 |
Answer: x 1 = 1, x 2 = 1, x 3 = 1
Implementation of the Gaussian method
The Gaussian method is implemented in many programming languages, in particular: Pascal, C++, php, Delphi, and there is also an online implementation of the Gaussian method.Using the Gaussian method
Application of the Gauss method in game theory
In game theory, when finding the maximin optimal strategy of a player, a system of equations is compiled, which is solved by the Gaussian method.Application of the Gauss method in solving differential equations
To find a partial solution to a differential equation, first find derivatives of the appropriate degree for the written partial solution (y=f(A,B,C,D)), which are substituted into the original equation. Next to find variables A,B,C,D a system of equations is compiled and solved by the Gaussian method.Application of the Jordano-Gauss method in linear programming
In linear programming, in particular in the simplex method, the rectangle rule, which uses the Jordano-Gauss method, is used to transform the simplex table at each iteration.Examples
Example No. 1. Solve the system using the Gaussian method:x 1 +2x 2 - 3x 3 + x 4 = -2
x 1 +2x 2 - x 3 + 2x 4 = 1
3x 1 -x 2 + 2x 3 + x 4 = 3
3x 1 +x 2 + x 3 + 3x 4 = 2
For ease of calculation, let's swap the lines:
Multiply the 2nd line by (-1). Add the 2nd line to the 1st
For ease of calculation, let's swap the lines:
From the 1st line we express x 4
From the 2nd line we express x 3
From the 3rd line we express x 2
From the 4th line we express x 1
Example No. 3.
- Solve SLAE using the Jordano-Gauss method. Let us write the system in the form: The resolving element is equal to (2.2). In place of the resolving element we get 1, and in the column itself we write zeros. All other elements of the matrix, including elements of column B, are determined by the rectangle rule. x 1 = 1.00, x 2 = 1.00, x 3 = 1.00
- Solve a system of linear equations using the Gauss method
ExampleSee how quickly you can tell if a system is collaborative
Video instruction
- Using the Gaussian method of eliminating unknowns, solve the system of linear equations. Check the solution found: Solution
- Solve a system of equations using the Gauss method. It is recommended that transformations associated with the sequential elimination of unknowns be applied to the extended matrix of a given system. Check the resulting solution.
Solution:xls - Solve a system of linear equations in three ways: a) the Gauss method of successive elimination of unknowns; b) according to the formula x = A -1 b with calculation inverse matrix A-1; c) according to Cramer's formulas.
Solution:xls - Solve the following degenerate system of equations using the Gauss method.
Download solution doc - Solve using the Gauss method a system of linear equations written in matrix form:
7 8 -3 x 92
2 2 2 y = 30
-9 -10 5 z -114
Solving a system of equations using the addition method
Solve the 6x+5y=3, 3x+3y=4 system of equations using the addition method.Solution.
6x+5y=3
3x+3y=4
Let's multiply the second equation by (-2).
6x+5y=3
-6x-6y=-8
============ (add)
-y=-5
Where does y = 5 come from?
Find x:
6x+5*5=3 or 6x=-22
Where does x = -22/6 = -11/3
Example No. 2. Solving an SLAE in matrix form means that the original record of the system must be reduced to a matrix record (the so-called extended matrix). Let's show this with an example.
Let's write the system in the form of an extended matrix:
|
|
|
|
|
|
|
|
x 3 = -21/(-21) = 1
x 2 = /15
x 1 = /3
From the 2nd line we express x 2:
From the 3rd line we express x 1:
Example No. 3. Solve the system using the Gaussian method: x 1 +2x 2 - 3x 3 + x 4 = -2
x 1 +2x 2 - x 3 + 2x 4 = 1
3x 1 -x 2 + 2x 3 + x 4 = 3
3x 1 +x 2 + x 3 + 3x 4 = 2
Solution:
Let's write the system in the form:
For ease of calculation, let's swap the lines:
Multiply the 2nd line by (-1). Add the 2nd line to the 1st
Let's multiply the 2nd line by (3). Multiply the 3rd line by (-1). Add the 3rd line to the 2nd
Multiply the 4th line by (-1). Add the 4th line to the 3rd
For ease of calculation, let's swap the lines:
Multiply the 1st line by (0). Add the 2nd line to the 1st
Multiply the 2nd line by (7). Let's multiply the 3rd line by (2). Add the 3rd line to the 2nd
Let's multiply the 1st line by (15). Let's multiply the 2nd line by (2). Add the 2nd line to the 1st
From the 1st line we express x 4
From the 2nd line we express x 3
From the 3rd line we express x 2
From the 4th line we express x 1
One of the simplest ways to solve a system of linear equations is a technique based on the calculation of determinants ( Cramer's rule). Its advantage is that it allows you to immediately record the solution; it is especially convenient in cases where the coefficients of the system are not numbers, but some parameters. Its disadvantage is the cumbersomeness of calculations in the case a large number equations; moreover, Cramer's rule is not directly applicable to systems in which the number of equations does not coincide with the number of unknowns. In such cases, it is usually used Gaussian method.
Systems of linear equations having the same set of solutions are called equivalent. Obviously, many solutions linear system does not change if any equations are swapped, or if one of the equations is multiplied by some non-zero number, or if one equation is added to another.
Gauss method (method of sequential elimination of unknowns) is that with the help of elementary transformations the system is reduced to an equivalent system of a step type. First, using the 1st equation, we eliminate x 1 of all subsequent equations of the system. Then, using the 2nd equation, we eliminate x 2 from the 3rd and all subsequent equations. This process, called direct Gaussian method, continues until there is only one unknown left on the left side of the last equation x n. After this it is done inverse of the Gaussian method– solving the last equation, we find x n; after that, using this value, from the penultimate equation we calculate x n–1, etc. We find the last one x 1 from the first equation.
It is convenient to carry out Gaussian transformations by performing transformations not with the equations themselves, but with the matrices of their coefficients. Consider the matrix:
called extended matrix of the system, because, in addition to the main matrix of the system, it includes a column of free terms. The Gaussian method is based on reducing the main matrix of the system to a triangular form (or trapezoidal form in the case of non-square systems) using elementary row transformations (!) of the extended matrix of the system.
Example 5.1. Solve the system using the Gaussian method:
Solution. Let's write out the extended matrix of the system and, using the first row, after that we will reset the remaining elements:
we get zeros in the 2nd, 3rd and 4th rows of the first column:
Now we need all elements in the second column below the 2nd row to be equal to zero. To do this, you can multiply the second line by –4/7 and add it to the 3rd line. However, in order not to deal with fractions, let's create a unit in the 2nd row of the second column and only
Now, to get a triangular matrix, you need to reset the element of the fourth row of the 3rd column; to do this, you can multiply the third row by 8/54 and add it to the fourth. However, in order not to deal with fractions, we will swap the 3rd and 4th rows and the 3rd and 4th columns and only after that we will reset the specified element. Note that when rearranging the columns, the corresponding variables change places and this must be remembered; other elementary transformations with columns (addition and multiplication by a number) cannot be performed!
The last simplified matrix corresponds to a system of equations equivalent to the original one:
From here, using the inverse of the Gaussian method, we find from the fourth equation x 3 = –1; from the third x 4 = –2, from the second x 2 = 2 and from the first equation x 1 = 1. In matrix form, the answer is written as
We considered the case when the system is definite, i.e. when there is only one solution. Let's see what happens if the system is inconsistent or uncertain.
Example 5.2. Explore the system using the Gaussian method:
Solution. We write out and transform the extended matrix of the system
We write a simplified system of equations:
Here, in the last equation it turns out that 0=4, i.e. contradiction. Consequently, the system has no solution, i.e. she incompatible. à
Example 5.3. Explore and solve the system using the Gaussian method:
Solution. We write out and transform the extended matrix of the system:
As a result of the transformations, the last line contains only zeros. This means that the number of equations has decreased by one:
Thus, after simplifications, there are two equations left, and four unknowns, i.e. two unknown "extra". Let them be "superfluous", or, as they say, free variables, will x 3 and x 4 . Then
Believing x 3 = 2a And x 4 = b, we get x 2 = 1–a And x 1 = 2b–a; or in matrix form
A solution written in this way is called general, because, giving parameters a And b different meanings, it is possible to describe all possible solutions of the system. a
In this article we:
- Let's define the Gaussian method,
- Let's analyze the algorithm of actions for solving linear equations, where the number of equations coincides with the number of unknown variables, and the determinant is not equal to zero;
- Let us analyze the algorithm of actions for solving SLAEs with a rectangular or singular matrix.
Gaussian method - what is it?
Definition 1Gauss method is a method that is used in solving systems of linear algebraic equations and has the following advantages:
- there is no need to check the system of equations for consistency;
- It is possible to solve systems of equations where:
- the number of determinants coincides with the number of unknown variables;
- the number of determinants does not coincide with the number of unknown variables;
- the determinant is zero.
- the result is produced with a relatively small number of computational operations.
Basic definitions and notations
Example 1There is a system of p linear equations with n unknowns (p can be equal to n):
a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 ⋯ a p 1 x 1 + a p 2 x 2 + . . . + a p n x n = b p ,
where x 1 , x 2 , . . . . , x n - unknown variables, a i j, i = 1, 2. . . , p , j = 1 , 2 . . . , n - numbers (real or complex), b 1 , b 2 , . . . , b n - free terms.
Definition 2
If b 1 = b 2 = . . . = b n = 0, then such a system of linear equations is called homogeneous, if vice versa - heterogeneous.
Definition 3
SLAE solution - set of values of unknown variables x 1 = a 1, x 2 = a 2, . . . , x n = a n , at which all equations of the system become identical to each other.
Definition 4
Joint SLAU - a system for which there is at least one solution option. Otherwise, it is called inconsistent.
Definition 5
Defined SLAU - This is a system that has a unique solution. If there is more than one solution, then such a system will be called uncertain.
Definition 6
Coordinate type of record:
a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 ⋯ a p 1 x 1 + a p 2 x 2 + . . . + a p n x n = b p
Definition 7
Matrix notation: A X = B, where
A = a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ a p 1 a p 2 ⋯ a p n - the main matrix of the SLAE;
X = x 1 x 2 ⋮ x n - column matrix of unknown variables;
B = b 1 b 2 ⋮ b n - matrix of free terms.
Definition 8
Extended Matrix - a matrix that is obtained by adding a matrix-column of free terms as an (n + 1) column and is designated T.
T = a 11 a 12 ⋮ a 1 n b 1 a 21 a 22 ⋮ a 2 n b 2 ⋮ ⋮ ⋮ ⋮ ⋮ a p 1 a p 2 ⋮ a p n b n
Definition 9
Singular square matrix A - a matrix whose determinant is equal to zero. If the determinant is not equal to zero, then such a matrix is then called non-degenerate.
Description of the algorithm for using the Gaussian method to solve SLAEs with an equal number of equations and unknowns (reverse and forward progression of the Gaussian method)
First, let's look at the definitions of forward and backward moves of the Gaussian method.
Definition 10
Forward Gaussian move - the process of sequential elimination of unknowns.
Definition 11
Gaussian reversal - the process of sequentially finding unknowns from the last equation to the first.
Gauss method algorithm:
Example 2
We solve a system of n linear equations with n unknown variables:
a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + . . . + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + . . . + a 3 n x n = b 3 ⋯ a n 1 x 1 + a n 2 x 2 + a n 3 x 3 + . . . + a n n x n = b n
Matrix determinant not equal to zero .
- a 11 is not equal to zero - this can always be achieved by rearranging the equations of the system;
- we exclude the variable x 1 from all equations of the system, starting from the second;
- Let's add to the second equation of the system the first one, which is multiplied by - a 21 a 11, add to the third equation the first one multiplied by - a 21 a 11, etc.
After these steps, the matrix will take the form:
a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n = b 1 a (1) 22 x 2 + a (1) 23 x 3 + . . . + a (1) 2 n x n = b (1) 2 a (1) 32 x 2 + a (1) 33 x 3 + . . . + a (1) 3 n x n = b (1) 3 ⋯ a (1) n 2 x 2 + a (1) n 3 x 3 + . . . + a (1) n n x n = b (1) n ,
where a i j (1) = a i j + a 1 j (- a i 1 a 11) , i = 2 , 3 , . . . , n , j = 2 , 3 , . . . , n , b i (1) = b i + b 1 (- a i 1 a 11) , i = 2 , 3 , . . . , n.
a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n = b 1 a (1) 22 x 2 + a (1) 23 x 3 + . . . + a (1) 2 n x n = b (1) 2 a (1) 32 x 2 + a (1) 33 x 3 + . . . + a (1) 3 n x n = b (1) 3 ⋯ a (1) n 2 x 2 + a (1) n 3 x 3 + . . . + a (1) n n x n = b (1) n
It is considered that a 22 (1) is not equal to zero. Thus, we proceed to eliminating the unknown variable x 2 from all equations, starting with the third:
- to the third equation of the system we add the second, which is multiplied by - a (1) 42 a (1) 22 ;
- to the fourth we add the second, which is multiplied by - a (1) 42 a (1) 22, etc.
After such manipulations, the SLAE has next view :
a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n = b 1 a (1) 22 x 2 + a (1) 23 x 3 + . . . + a (1) 2 n x n = b (1) 2 a (2) 33 x 3 + . . . + a (2) 3 n x n = b (2) 3 ⋯ a (2) n 3 x 3 + . . . + a (2) n n x n = b (2) n ,
where a i j (2) = a (1) i j + a 2 j (- a (1) i 2 a (1) 22) , i = 3 , 4 , . . . , n , j = 3 , 4 , . . . , n , b i (2) = b (1) i + b (1) 2 (- a (1) i 2 a (1) 22) , i = 3 , 4 , . . . , n. .
Thus, the variable x 2 is excluded from all equations, starting from the third.
a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n = b 1 a (1) 22 x 2 + a (1) 23 x 3 + . . . + a (1) 2 n x n = b (1) 2 a (2) 33 x 3 + . . . + a (2) 3 n x n = b (2) 3 ⋯ a (n - 1) n n x n = b (n - 1) n
Note
After the system has taken this form, you can start inverse of the Gaussian method :
- calculate x n from the last equation as x n = b n (n - 1) a n n (n - 1) ;
- using the resulting x n, we find x n - 1 from the penultimate equation, etc., find x 1 from the first equation.
Example 3
Find a solution to the system of equations using the Gauss method:
How to decide?
The coefficient a 11 is different from zero, so we proceed to the direct solution, i.e. to the exclusion of the variable x 11 from all equations of the system except the first. In order to do this, we add to the left and right sides of the 2nd, 3rd and 4th equations the left and right sides of the first, which are multiplied by - a 21 a 11:
1 3, - a 31 a 11 = - - 2 3 = 2 3 and - a 41 a 11 = - 1 3.
3 x 1 + 2 x 2 + x 3 + x 4 = - 2 x 1 - x 2 + 4 x 3 - x 4 = - 1 - 2 x 1 - 2 x 2 - 3 x 3 + x 4 = 9 x 1 + 5 x 2 - x 3 + 2 x 4 = 4 ⇔
⇔ 3 x 1 + 2 x 2 + x 3 + x 4 = - 2 x 1 - x 2 + 4 x 3 - x 4 + (- 1 3) (3 x 1 + 2 x 2 + x 3 + x 4) = - 1 + (- 1 3) (- 2) - 2 x 1 - 2 x 2 - 3 x 3 + x 4 + 2 3 (3 x 1 + 2 x 2 + x 3 + x 4) = 9 + 2 3 (- 2) x 1 + 5 x 2 - x 3 + 2 x 4 + (- 1 3) (3 x 1 + 2 x 2 + x 3 + x 4) = 4 + (- 1 3) (- 2 ) ⇔
⇔ 3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 2 3 x 2 - 7 3 x 3 + 5 3 x 4 = 23 3 13 3 x 2 - 4 3 x 3 + 5 3 x 4 = 14 3
We have eliminated the unknown variable x 1, now we proceed to eliminate the variable x 2:
A 32 (1) a 22 (1) = - - 2 3 - 5 3 = - 2 5 and a 42 (1) a 22 (1) = - 13 3 - 5 3 = 13 5:
3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 2 3 x 2 - 7 3 x 3 + 5 3 x 4 = 23 3 13 3 x 2 - 4 3 x 3 + 5 3 x 4 = 14 3 ⇔
⇔ 3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 2 3 x 2 - 7 3 x 3 + 5 3 x 4 + (- 2 5) (- 5 3 x 2 + 11 3 x 3 - 4 3 x 4) = 23 3 + (- 2 5) (- 1 3) 13 3 x 2 - 4 3 x 3 + 5 3 x 4 + 13 5 (- 5 3 x 2 + 11 3 x 3 - 4 3 x 4) = 14 3 + 13 5 (- 1 3) ⇔
⇔ 3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 19 5 x 3 + 11 5 x 4 = 39 5 41 5 x 3 - 9 5 x 4 = 19 5
In order to complete the forward progression of the Gaussian method, it is necessary to exclude x 3 from the last equation of the system - a 43 (2) a 33 (2) = - 41 5 - 19 5 = 41 19:
3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 19 5 x 3 + 11 5 x 4 = 39 5 41 5 x 3 - 9 5 x 4 = 19 5 ⇔
3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 19 5 x 3 + 11 5 x 4 = 39 5 41 5 x 3 - 9 5 x 4 + 41 19 (- 19 5 x 3 + 11 5 x 4) = 19 5 + 41 19 39 5 ⇔
⇔ 3 x 1 + 2 x 2 + x 3 + x 4 = - 2 - 5 3 x 2 + 11 3 x 3 - 4 3 x 4 = - 1 3 - 19 5 x 3 + 11 5 x 4 = 39 5 56 19 x 4 = 392 19
The reverse of the Gauss method:
- from the last equation we have: x 4 \u003d 392 19 56 19 \u003d 7;
- from the 3rd equation we get: x 3 \u003d - 5 19 (39 5 - 11 5 x 4) \u003d - 5 19 (39 5 - 11 5 × 7) \u003d 38 19 \u003d 2;
- from the 2nd: x 2 \u003d - 3 5 (- 1 3 - 11 3 x 4 + 4 3 x 4) \u003d - 3 5 (- 1 3 - 11 3 × 2 + 4 3 × 7) \u003d - 1;
- from the 1st: x 1 \u003d 1 3 (- 2 - 2 x 2 - x 3 - x 4) \u003d - 2 - 2 × (- 1) - 2 - 7 3 \u003d - 9 3 \u003d - 3.
Answer : x 1 = - 3 ; x 2 = - 1 ; x 3 = 2 ; x 4 = 7
Example 4
Find a solution to the same example using the Gaussian method in matrix notation:
3 x 1 + 2 x 2 + x 3 + x 4 = - 2 x 1 - x 2 + 4 x 3 - x 4 = - 1 - 2 x 1 - 2 x 2 - 3 x 3 + x 4 = 9 x 1 + 5 x 2 - x 3 + 2 x 4 = 4
How to decide?
The extended matrix of the system is presented as:
x 1 x 2 x 3 x 4 3 2 1 1 1 - 1 4 - 1 - 2 - 2 - 3 1 1 5 - 1 2 - 2 - 1 9 4
The direct approach of the Gaussian method in this case involves reducing the extended matrix to a trapezoidal form using elementary transformations. This process is very similar to the process of eliminating unknown variables in coordinate form.
Matrix transformation begins with turning all elements zero. To do this, to the elements of the 2nd, 3rd and 4th lines we add the corresponding elements of the 1st line, which are multiplied by - a 21 a 11 = - 1 3 , - a 31 a 11 = - - 2 3 = 2 3 i n a - a 41 a 11 = - 1 3 .
Further transformations occur according to the following scheme: all elements in the 2nd column, starting from the 3rd row, become zero. Such a process corresponds to the process of eliminating a variable. In order to perform this action, it is necessary to add to the elements of the 3rd and 4th rows the corresponding elements of the 1st row of the matrix, which is multiplied by - a 32 (1) a 22 (1) = - 2 3 - 5 3 = - 2 5 and - a 42 (1) a 22 (1) \u003d - 13 3 - 5 3 \u003d 13 5:
x 1 x 2 x 3 x 4 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 - 2 3 - 7 3 5 3 | 23 3 0 13 3 - 4 3 5 3 | 14 3 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 - 2 3 + (- 2 5) (- 5 3) - 7 3 + (- 2 5) 11 3 5 3 + (- 2 5) (- 4 3) | 23 3 + (- 2 5) (- 1 3) 0 13 3 + 13 5 (- 5 3) - 4 3 + 13 5 × 11 3 5 3 + 13 5 (- 4 3) | 14 3 + 13 5 (- 1 3) ~
x 1 x 2 x 3 x 4 ~ 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 41 5 - 9 5 | 19 5
Now we exclude the variable x 3 from the last equation - we add to the elements of the last row of the matrix the corresponding elements of the last row, which is multiplied by a 43 (2) a 33 (2) = - 41 5 - 19 5 = 41 19.
x 1 x 2 x 3 x 4 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 41 5 - 9 5 | 19 5 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 41 5 + 41 19 (- 19 5) - 9 5 + 41 19 × 11 5 | 19 5 + 41 19 × 39 5 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 0 56 19 | 392 19
Now let's apply the reverse method. In matrix notation, the transformation of the matrix is such that the matrix, which is marked in color in the image:
x 1 x 2 x 3 x 4 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 0 56 19 | 392 19
became diagonal, i.e. took the following form:
x 1 x 2 x 3 x 4 3 0 0 0 | a 1 0 - 5 3 0 0 | a 2 0 0 - 19 5 0 | a 3 0 0 0 56 19 | 392 19, where a 1, a 2, and 3 are some numbers.
Such transformations are analogous to the forward motion, only the transformations are performed not from the 1st line of the equation, but from the last. We add to the elements of the 3rd, 2nd and 1st lines the corresponding elements of the last line, which is multiplied by
11 5 56 19 = - 209 280, on - - 4 3 56 19 = 19 42 and on - 1 56 19 = 19 56.
x 1 x 2 x 3 x 4 3 2 1 1 | - 2 0 - 5 3 11 3 - 4 3 | - 1 3 0 0 - 19 5 11 5 | 39 5 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 1 + (- 19 56) 56 19 | - 2 + (- 19 56) 392 19 0 - 5 3 11 3 - 4 3 + 19 42 × 56 19 | - 1 3 + 19 42 × 392 19 0 0 - 19 5 11 5 + (- 209 280) 56 19 | 39 5 + (- 209 280) 392 19 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 0 | - 9 0 - 5 3 11 3 0 | 9 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19
11 3 - 19 5 = 55 57 and on - 1 - 19 5 = 5 19.
x 1 x 2 x 3 x 4 3 2 1 0 | - 9 0 - 5 3 11 3 0 | 9 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 + 5 19 (- 19 5) 0 | - 9 + 5 19 (- 38 5) 0 - 5 3 11 3 + 55 57 (- 19 5) 0 | 9 + 55 57 (- 38 5) 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 2 1 0 | - 11 0 - 5 3 0 0 | 5 3 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19
At the last stage, we add the elements of the 2nd row to the corresponding elements of the 1st row, which are multiplied by - 2 - 5 3 = 6 5 .
x 1 x 2 x 3 x 4 3 2 1 0 | - 11 0 - 5 3 0 0 | 5 3 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 2 + 6 5 (- 5 3) 0 0 | - 11 + 6 5 × 5 3) 0 - 5 3 0 0 | 5 3 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19 ~
x 1 x 2 x 3 x 4 ~ 3 0 0 0 | - 9 0 - 5 3 0 0 | 5 3 0 0 - 19 5 0 | - 38 5 0 0 0 56 19 | 392 19
The resulting matrix corresponds to the system of equations
3 x 1 = - 9 - 5 3 x 2 = 5 3 - 19 5 x 3 = - 38 5 56 19 x 4 = 392 19, from where we find the unknown variables.
Answer: x 1 = - 3, x 2 = - 1, x 3 = 2, x 4 = 7.
Description of the algorithm for using the Gauss method for solving SLAEs with a divergent number of equations and unknowns, or with a degenerate matrix system
Definition 2If the underlying matrix is square or rectangular, then systems of equations may have a unique solution, may not have solutions, or may have an infinite number of solutions.
From this section we will learn how to use the Gaussian method to determine the compatibility or incompatibility of SLAEs, and also, in the case of compatibility, determine the number of solutions for the system.
In principle, the method of eliminating unknowns for such SLAEs remains the same, but there are several points that need to be emphasized.
Example 5
At some stages of eliminating unknowns, some equations turn into identities 0=0. In this case, the equations can be safely removed from the system and the direct progression of the Gaussian method can be continued.
If we exclude x 1 from the 2nd and 3rd equations, then the situation turns out to be as follows:
x 1 + 2 x 2 - x 3 + 3 x 4 = 7 2 x 1 + 4 x 2 - 2 x 3 + 6 x 4 = 14 x - x + 3 x + x = - 1 ⇔
x 1 + 2 x 2 - x 3 + 3 x 4 = 7 2 x 1 + 4 x 2 - 2 x 3 + 6 x 4 + (- 2) (x 1 + 2 x 2 - x 3 + 3 x 4) = 14 + (- 2) × 7 x - x + 3 x + x + (- 1) (x 1 + 2 x 2 - x 3 + 3 x 4) = - 1 + (- 1) × 7 ⇔
⇔ x 1 + 2 x 2 - x 3 + 3 x 4 = 7 0 = 0 - 3 x 2 + 4 x 3 - 2 x 4 = - 8
It follows from this that the 2nd equation can be safely removed from the system and the solution can be continued.
If we carry out the direct progression of the Gaussian method, then one or more equations can take the form of a certain number that is different from zero.
This indicates that the equation that turns into equality 0 = λ cannot turn into equality for any values of the variables. Simply put, such a system is inconsistent (has no solution).
Result:
- If, when carrying out the forward progression of the Gaussian method, one or more equations take the form 0 = λ, where λ is a certain number that is different from zero, then the system is inconsistent.
- If, at the end of the forward run of the Gaussian method, a system is obtained whose number of equations coincides with the number of unknowns, then such a system is consistent and defined: it has a unique solution, which is calculated by the reverse run of the Gaussian method.
- If, at the end of the forward run of the Gaussian method, the number of equations in the system turns out to be less than the number of unknowns, then such a system is consistent and has an infinite number of solutions, which are calculated during the reverse run of the Gaussian method.
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