Gauss method applied. Gauss method for solving matrices
Ever since the beginning of the 16th-18th centuries, mathematicians began to intensively study the functions, thanks to which so much has changed in our lives. Computer technology without this knowledge would simply not exist. For solutions challenging tasks, linear equations and functions were created various concepts, theorems and methods of solution. One of such universal and rational ways and methods of solving linear equations and their systems became the Gaussian method. Matrices, their rank, determinant - everything can be calculated without using complex operations.
What is SLAU
In mathematics, there is the concept of SLAE - a system of linear algebraic equations. What does she represent? This is a set of m equations with the required n unknowns, usually denoted as x, y, z, or x 1 , x 2 ... x n, or other symbols. Solve by Gauss method this system- means to find all the required unknowns. If the system has the same number unknowns and equations, then it is called an n-th order system.
The most popular methods for solving SLAE
V educational institutions secondary education are studying various techniques for solving such systems. Most often this simple equations, consisting of two unknowns, so any existing method it won't take long to find answers to them. It can be like a substitution method, when another equation is derived from one equation and substituted into the original one. Or term by term subtraction and addition. But the Gauss method is considered the easiest and most universal. It makes it possible to solve equations with any number of unknowns. Why is this technique considered rational? Everything is simple. The matrix method is good because it does not require several times to rewrite unnecessary characters in the form of unknowns, it is enough to do arithmetic operations on the coefficients - and you will get a reliable result.
Where are SLAEs used in practice?
The solution of SLAE are the points of intersection of lines on the graphs of functions. In our high-tech computer age, people who are closely involved in the development of games and other programs need to know how to solve such systems, what they represent and how to check the correctness of the resulting result. Most often, programmers develop special linear algebra calculators, this includes a system of linear equations. The Gauss method allows you to calculate all existing solutions. Other simplified formulas and techniques are also used.
SLAE compatibility criterion
Such a system can only be solved if it is compatible. For clarity, we present the SLAE in the form Ax=b. It has a solution if rang(A) equals rang(A,b). In this case, (A,b) is an extended form matrix that can be obtained from matrix A by rewriting it with free terms. It turns out that solving linear equations using the Gaussian method is quite easy.
Perhaps some notation is not entirely clear, so it is necessary to consider everything with an example. Let's say there is a system: x+y=1; 2x-3y=6. It consists of only two equations in which there are 2 unknowns. The system will have a solution only if the rank of its matrix is equal to the rank of the augmented matrix. What is a rank? This is the number of independent lines of the system. In our case, the rank of the matrix is 2. Matrix A will consist of the coefficients located near the unknowns, and the coefficients behind the “=” sign will also fit into the expanded matrix.
Why SLAE can be represented in matrix form
Based on the compatibility criterion according to the proven Kronecker-Capelli theorem, the system of linear algebraic equations can be represented in matrix form. Using the Gaussian cascade method, you can solve the matrix and get the only reliable answer for the entire system. If the rank of an ordinary matrix is equal to the rank of its extended matrix, but less than the number of unknowns, then the system has an infinite number of answers.
Matrix transformations
Before moving on to solving matrices, it is necessary to know what actions can be performed on their elements. There are several elementary transformations:
- By rewriting the system into a matrix form and carrying out its solution, it is possible to multiply all the elements of the series by the same coefficient.
- In order to convert a matrix to canonical form, two parallel rows can be swapped. The canonical form implies that all elements of the matrix that are located along the main diagonal become ones, and the remaining ones become zeros.
- The corresponding elements of the parallel rows of the matrix can be added one to the other.
Jordan-Gauss method
The essence of solving systems of linear homogeneous and inhomogeneous equations Gaussian method is to gradually eliminate the unknowns. Let's say we have a system of two equations in which there are two unknowns. To find them, you need to check the system for compatibility. The Gaussian equation is solved very simply. It is necessary to write out the coefficients located near each unknown in a matrix form. To solve the system, you need to write out the augmented matrix. If one of the equations contains a smaller number of unknowns, then "0" must be put in place of the missing element. All known transformation methods are applied to the matrix: multiplication, division by a number, adding the corresponding elements of the rows to each other, and others. It turns out that in each row it is necessary to leave one variable with the value "1", the rest should be reduced to zero. For a more accurate understanding, it is necessary to consider the Gauss method with examples.
A simple example of solving a 2x2 system
To begin with, let's take a simple system of algebraic equations, in which there will be 2 unknowns.
Let's rewrite it in an augmented matrix.
To solve this system of linear equations, only two operations are required. We need to bring the matrix to the canonical form so that there are units along the main diagonal. So, translating from the matrix form back into the system, we get the equations: 1x+0y=b1 and 0x+1y=b2, where b1 and b2 are the answers obtained in the process of solving.
- The first step in solving the augmented matrix will be as follows: the first row must be multiplied by -7 and the corresponding elements added to the second row, respectively, in order to get rid of one unknown in the second equation.
- Since the solution of equations by the Gauss method implies bringing the matrix to the canonical form, then it is necessary to do the same operations with the first equation and remove the second variable. To do this, we subtract the second line from the first and get the necessary answer - the solution of the SLAE. Or, as shown in the figure, we multiply the second row by a factor of -1 and add the elements of the second row to the first row. This is the same.
As you can see, our system is solved by the Jordan-Gauss method. We rewrite it in the required form: x=-5, y=7.
An example of solving SLAE 3x3
Suppose we have a more complex system of linear equations. The Gauss method makes it possible to calculate the answer even for the most seemingly confusing system. Therefore, in order to delve deeper into the calculation methodology, we can move on to a more complex example with three unknowns.
As in the previous example, we rewrite the system in the form of an expanded matrix and begin to bring it to the canonical form.
To solve this system, you will need to perform much more actions than in the previous example.
- First you need to make in the first column one single element and the rest zeros. To do this, multiply the first equation by -1 and add the second equation to it. It is important to remember that we rewrite the first line in original form, and the second - already in the modified.
- Next, we remove the same first unknown from the third equation. To do this, we multiply the elements of the first row by -2 and add them to the third row. Now the first and second lines are rewritten in their original form, and the third - already with changes. As you can see from the result, we got the first one at the beginning of the main diagonal of the matrix and the rest are zeros. A few more actions, and the system of equations by the Gauss method will be reliably solved.
- Now you need to do operations on other elements of the rows. The third and fourth steps can be combined into one. We need to divide the second and third lines by -1 to get rid of the negative ones on the diagonal. We have already brought the third line to the required form.
- Next, we canonicalize the second line. To do this, we multiply the elements of the third row by -3 and add them to the second line of the matrix. It can be seen from the result that the second line is also reduced to the form we need. It remains to do a few more operations and remove the coefficients of the unknowns from the first row.
- In order to make 0 from the second element of the row, you need to multiply the third row by -3 and add it to the first row.
- The next decisive step is to add to the first line necessary elements second row. So we get the canonical form of the matrix, and, accordingly, the answer.
As you can see, the solution of equations by the Gauss method is quite simple.
An example of solving a 4x4 system of equations
Some more complex systems of equations can be solved by the Gaussian method using computer programs. It is necessary to drive coefficients for unknowns into existing empty cells, and the program will calculate the required result step by step, describing each action in detail.
Described below step-by-step instruction solutions to this example.
In the first step, free coefficients and numbers for unknowns are entered into empty cells. Thus, we get the same augmented matrix that we write by hand.
And all the necessary arithmetic operations are performed to bring the extended matrix to the canonical form. It must be understood that the answer to a system of equations is not always integers. Sometimes the solution can be from fractional numbers.
Checking the correctness of the solution
The Jordan-Gauss method provides for checking the correctness of the result. In order to find out whether the coefficients are calculated correctly, you just need to substitute the result into the original system of equations. The left side of the equation must match the right side, which is behind the equals sign. If the answers do not match, then you need to recalculate the system or try to apply another method of solving SLAE known to you, such as substitution or term-by-term subtraction and addition. After all, mathematics is a science that has a huge number of various techniques solutions. But remember: the result should always be the same, no matter what solution method you used.
Gauss method: the most common errors in solving SLAE
During the solution of linear systems of equations, errors most often occur, such as incorrect transfer of coefficients to a matrix form. There are systems in which some unknowns are missing in one of the equations, then, transferring the data to the expanded matrix, they can be lost. As a result, when solving this system, the result may not correspond to the real one.
Another of the main mistakes can be incorrect writing out the final result. It must be clearly understood that the first coefficient will correspond to the first unknown from the system, the second - to the second, and so on.
The Gauss method describes in detail the solution of linear equations. Thanks to him, it is easy to perform the necessary operations and find the right result. In addition, this is a universal tool for finding a reliable answer to equations of any complexity. Maybe that is why it is so often used in solving SLAE.
We continue to consider systems of linear equations. This lesson is the third on the topic. If you have a vague idea of what a system of linear equations is in general, you feel like a teapot, then I recommend starting with the basics on the Next page, it is useful to study the lesson.
Gauss method is easy! Why? The famous German mathematician Johann Carl Friedrich Gauss, during his lifetime, received recognition as the greatest mathematician of all time, a genius, and even the nickname "King of Mathematics". And everything ingenious, as you know, is simple! By the way, not only suckers, but also geniuses get into the money - the portrait of Gauss flaunted on a bill of 10 Deutschmarks (before the introduction of the euro), and Gauss still mysteriously smiles at the Germans from ordinary postage stamps.
The Gauss method is simple in that it IS ENOUGH THE KNOWLEDGE OF A FIFTH-GRADE STUDENT to master it. Must be able to add and multiply! It is no coincidence that the method of successive elimination of unknowns is often considered by teachers at school mathematical electives. It is a paradox, but the Gauss method causes the greatest difficulties for students. Nothing surprising - it's all about the methodology, and I will try to tell in an accessible form about the algorithm of the method.
First, we systematize the knowledge about systems of linear equations a little. A system of linear equations can:
1) Have a unique solution. 2) Have infinitely many solutions. 3) Have no solutions (be incompatible).
The Gauss method is the most powerful and versatile tool for finding a solution any systems of linear equations. As we remember Cramer's rule and matrix method are unsuitable in cases where the system has infinitely many solutions or is inconsistent. A method of successive elimination of unknowns anyway lead us to the answer! In this lesson, we will again consider the Gauss method for case No. 1 (the only solution to the system), an article is reserved for the situations of points No. 2-3. I note that the method algorithm itself works in the same way in all three cases.
Back to the simplest system from the lesson How to solve a system of linear equations? and solve it using the Gaussian method.
The first step is to write extended matrix system: . By what principle the coefficients are recorded, I think everyone can see. The vertical line inside the matrix does not carry any mathematical meaning - it's just a strikethrough for ease of design.
reference : I recommend to remember terms linear algebra. System Matrix is a matrix composed only of coefficients for unknowns, in this example, the matrix of the system: . Extended System Matrix is the same matrix of the system plus a column of free members, in this case: . Any of the matrices can be called simply a matrix for brevity.
After the extended matrix of the system is written, it is necessary to perform some actions with it, which are also called elementary transformations.
There are the following elementary transformations:
1) Strings matrices can rearrange places. For example, in the matrix under consideration, you can safely rearrange the first and second rows: 
2) If there are (or appeared) proportional (as a special case - identical) rows in the matrix, then it follows delete from the matrix, all these rows except one. Consider, for example, the matrix 
. In this matrix, the last three rows are proportional, so it is enough to leave only one of them: 
.
3) If a zero row appeared in the matrix during the transformations, then it also follows delete. I will not draw, of course, the zero line is the line in which only zeros.
4) The row of the matrix can be multiply (divide) for any number non-zero. Consider, for example, the matrix . Here it is advisable to divide the first line by -3, and multiply the second line by 2: 
. This action is very useful, as it simplifies further transformations of the matrix.
5) This transformation causes the most difficulties, but in fact there is nothing complicated either. To the row of the matrix, you can add another string multiplied by a number, different from zero. Consider our matrix from case study: . First, I will describe the transformation in great detail. Multiply the first row by -2: 
, and to the second line we add the first line multiplied by -2: 
. Now the first line can be divided "back" by -2: . As you can see, the line that is ADDED LI – hasn't changed. Is always the line is changed, TO WHICH ADDED UT.
In practice, of course, they don’t paint in such detail, but write shorter: 
Once again: to the second line added the first row multiplied by -2. The line is usually multiplied orally or on a draft, while the mental course of calculations is something like this:
“I rewrite the matrix and rewrite the first row: 
»
First column first. Below I need to get zero. Therefore, I multiply the unit above by -2:, and add the first to the second line: 2 + (-2) = 0. I write the result in the second line: 
»
“Now the second column. Above -1 times -2: . I add the first to the second line: 1 + 2 = 3. I write the result to the second line: 
»
“And the third column. Above -5 times -2: . I add the first line to the second line: -7 + 10 = 3. I write the result in the second line: 
»
Please think carefully about this example and understand the sequential calculation algorithm, if you understand this, then the Gauss method is practically "in your pocket". But, of course, we are still working on this transformation.
Elementary transformations do not change the solution of the system of equations
! ATTENTION: considered manipulations can not use, if you are offered a task where the matrices are given "by themselves". For example, with "classic" matrices in no case should you rearrange something inside the matrices! Let's return to our system. She's practically broken into pieces.
Let us write the augmented matrix of the system and, using elementary transformations, reduce it to stepped view:
(1) The first row was added to the second row, multiplied by -2. And again: why do we multiply the first row by -2? In order to get zero at the bottom, which means getting rid of one variable in the second line.
(2) Divide the second row by 3.
The purpose of elementary transformations
–
convert the matrix to step form: 
. In the design of the task, they directly draw out the “ladder” with a simple pencil, and also circle the numbers that are located on the “steps”. The term "stepped view" itself is not entirely theoretical; in the scientific and educational literature, it is often called trapezoidal view or triangular view.
As a result of elementary transformations, we have obtained equivalent original system of equations:
Now the system needs to be "untwisted" in the opposite direction - from the bottom up, this process is called reverse Gauss method.
In the lower equation, we already have the finished result: .
Consider the first equation of the system and substitute the already known value of “y” into it:
Let us consider the most common situation, when the Gaussian method is required to solve a system of three linear equations with three unknowns.
Example 1
Solve the system of equations using the Gauss method: 
Let's write the augmented matrix of the system: 
Now I will immediately draw the result that we will come to in the course of the solution: 
And I repeat, our goal is to bring the matrix to a stepped form using elementary transformations. Where to start taking action?
First, look at the top left number: 
Should almost always be here unit. Generally speaking, -1 (and sometimes other numbers) will also suit, but somehow it has traditionally happened that a unit is usually placed there. How to organize a unit? We look at the first column - we have a finished unit! Transformation one: swap the first and third lines: 
Now the first line will remain unchanged until the end of the solution. Now fine.
The unit in the top left is organized. Now you need to get zeros in these places: 
Zeros are obtained just with the help of a "difficult" transformation. First, we deal with the second line (2, -1, 3, 13). What needs to be done to get zero in the first position? Need to to the second line add the first line multiplied by -2. Mentally or on a draft, we multiply the first line by -2: (-2, -4, 2, -18). And we consistently carry out (again mentally or on a draft) addition, to the second line we add the first line, already multiplied by -2:
The result is written in the second line: 
Similarly, we deal with the third line (3, 2, -5, -1). To get zero in the first position, you need to the third line add the first line multiplied by -3. Mentally or on a draft, we multiply the first line by -3: (-3, -6, 3, -27). AND to the third line we add the first line multiplied by -3:
The result is written in the third line:
In practice, these actions are usually performed verbally and written down in one step: 
No need to count everything at once and at the same time. The order of calculations and "insertion" of results consistent and usually like this: first we rewrite the first line, and puff ourselves quietly - CONSISTENTLY and CAREFULLY:
And I have already considered the mental course of the calculations themselves above.
In this example, this is easy to do, we divide the second line by -5 (since all numbers there are divisible by 5 without a remainder). At the same time, we divide the third line by -2, because the smaller the number, the simpler the solution:
At the final stage of elementary transformations, one more zero must be obtained here: 
For this to the third line we add the second line, multiplied by -2:
Try to parse this action yourself - mentally multiply the second line by -2 and carry out the addition.
The last action performed is the hairstyle of the result, divide the third line by 3.
As a result of elementary transformations, an equivalent initial system of linear equations was obtained: 
Cool.
Now the reverse course of the Gaussian method comes into play. The equations "unwind" from the bottom up.
In the third equation, we already have the finished result:
Let's look at the second equation: . The meaning of "z" is already known, thus:
And finally, the first equation: . "Y" and "Z" are known, the matter is small:
Answer: 
As has been repeatedly noted, for any system of equations, it is possible and necessary to check the found solution, fortunately, this is not difficult and fast.
Example 2
This is an example for self-solving, a sample of finishing and an answer at the end of the lesson.
It should be noted that your course of action may not coincide with my course of action, and this is a feature of the Gauss method. But the answers must be the same!
Example 3
Solve a system of linear equations using the Gauss method 
We look at the upper left "step". There we should have a unit. The problem is that there are no ones in the first column at all, so nothing can be solved by rearranging the rows. In such cases, the unit must be organized using an elementary transformation. This can usually be done in several ways. I did this: (1) To the first line we add the second line, multiplied by -1. That is, we mentally multiplied the second line by -1 and performed the addition of the first and second lines, while the second line did not change.
Now at the top left "minus one", which suits us perfectly. Who wants to get +1 can perform an additional gesture: multiply the first line by -1 (change its sign).
(2) The first row multiplied by 5 was added to the second row. The first row multiplied by 3 was added to the third row.
(3) The first line was multiplied by -1, in principle, this is for beauty. The sign of the third line was also changed and moved to the second place, thus, on the second “step, we had the desired unit.
(4) The second line multiplied by 2 was added to the third line.
(5) The third row was divided by 3.
A bad sign that indicates a calculation error (less often a typo) is a “bad” bottom line. That is, if we got something like below, and, accordingly, 
, then with a high degree of probability it can be argued that an error was made in the course of elementary transformations.
We charge the reverse move, in the design of examples, the system itself is often not rewritten, and the equations are “taken directly from the given matrix”. The reverse move, I remind you, works from the bottom up. Yes, here is a gift:
Answer: 
.
Example 4
Solve a system of linear equations using the Gauss method 
This is an example for an independent solution, it is somewhat more complicated. It's okay if someone gets confused. Full solution and design sample at the end of the lesson. Your solution may differ from mine.
In the last part, we consider some features of the Gauss algorithm. The first feature is that sometimes some variables are missing in the equations of the system, for example: 
How to correctly write the augmented matrix of the system? I already talked about this moment in the lesson. Cramer's rule. Matrix method. In the expanded matrix of the system, we put zeros in place of the missing variables: 
By the way, this is a fairly easy example, since there is already one zero in the first column, and there are fewer elementary transformations to perform.
The second feature is this. In all the examples considered, we placed either –1 or +1 on the “steps”. Could there be other numbers? In some cases they can. Consider the system: 
.
Here on the upper left "step" we have a deuce. But we notice the fact that all the numbers in the first column are divisible by 2 without a remainder - and another two and six. And the deuce at the top left will suit us! At the first step, you need to perform the following transformations: add the first line multiplied by -1 to the second line; to the third line add the first line multiplied by -3. Thus, we will get the desired zeros in the first column.
Or else like this conditional example: 
. Here, the triple on the second “rung” also suits us, since 12 (the place where we need to get zero) is divisible by 3 without a remainder. It is necessary to carry out the following transformation: to the third line, add the second line, multiplied by -4, as a result of which the zero we need will be obtained.
The Gauss method is universal, but there is one peculiarity. Confidently learn to solve systems by other methods (Cramer's method, matrix method) can be literally the first time - there is a very strict algorithm. But in order to feel confident in the Gauss method, you should “fill your hand” and solve at least 5-10 ten systems. Therefore, at first there may be confusion, errors in calculations, and there is nothing unusual or tragic in this.
Rainy autumn weather outside the window .... Therefore, for everyone, a more complex example for an independent solution:
Example 5
Solve a system of 4 linear equations with four unknowns using the Gauss method. 
Such a task in practice is not so rare. I think that even a teapot who has studied this page in detail understands the algorithm for solving such a system intuitively. Basically the same - just more action.
The cases when the system has no solutions (inconsistent) or has infinitely many solutions are considered in the lesson. Incompatible systems and systems with a common solution. There you can fix the considered algorithm of the Gauss method.
Wish you success!
Solutions and answers:
Example 2:
Solution
:
Let us write down the extended matrix of the system and, using elementary transformations, bring it to a stepped form.
Performed elementary transformations:
(1) The first row was added to the second row, multiplied by -2. The first line was added to the third line, multiplied by -1.
Attention!
Here it may be tempting to subtract the first from the third line, I strongly do not recommend subtracting - the risk of error greatly increases. We just fold!
(2) The sign of the second line was changed (multiplied by -1). The second and third lines have been swapped.
note
that on the “steps” we are satisfied not only with one, but also with -1, which is even more convenient.
(3) To the third line, add the second line, multiplied by 5.
(4) The sign of the second line was changed (multiplied by -1). The third line was divided by 14.
Reverse move:
Answer
:
.
Example 4:
Solution
:
We write the extended matrix of the system and, using elementary transformations, bring it to a step form:
Conversions performed: (1) The second line was added to the first line. Thus, the desired unit is organized on the upper left “step”. (2) The first row multiplied by 7 was added to the second row. The first row multiplied by 6 was added to the third row.
With the second "step" everything is worse , the "candidates" for it are the numbers 17 and 23, and we need either one or -1. Transformations (3) and (4) will be aimed at obtaining the desired unit (3) The second line was added to the third line, multiplied by -1. (4) The third line, multiplied by -3, was added to the second line. The necessary thing on the second step is received . (5) To the third line added the second, multiplied by 6. (6) The second row was multiplied by -1, the third row was divided by -83.
Reverse move:
Answer :
Example 5:
Solution
:
Let us write down the matrix of the system and, using elementary transformations, bring it to a stepwise form:
Conversions performed: (1) The first and second lines have been swapped. (2) The first row was added to the second row, multiplied by -2. The first line was added to the third line, multiplied by -2. The first line was added to the fourth line, multiplied by -3. (3) The second line multiplied by 4 was added to the third line. The second line multiplied by -1 was added to the fourth line. (4) The sign of the second line has been changed. The fourth line was divided by 3 and placed instead of the third line. (5) The third line was added to the fourth line, multiplied by -5.
Reverse move:
Answer :
This online calculator finds a solution to a system of linear equations (SLE) using the Gaussian method. given detailed solution. To calculate, choose the number of variables and the number of equations. Then enter the data in the cells and click on the "Calculate."
|
Number representation:
Integers and (or) Common fractionsIntegers and/or Decimals
Number of digits after decimal separator
×
Warning
Clear all cells?
Close Clear
Data entry instruction. Numbers are entered as whole numbers (examples: 487, 5, -7623, etc.), decimal numbers (eg. 67., 102.54, etc.) or fractions. The fraction must be typed in the form a/b, where a and b (b>0) are integers or decimal numbers. Examples 45/5, 6.6/76.4, -7/6.7, etc.
Gauss method
The Gauss method is a method of transition from the original system of linear equations (using equivalent transformations) to a system that is easier to solve than the original system.
The equivalent transformations of the system of linear equations are:
- swapping two equations in the system,
- multiplication of any equation in the system by a non-zero real number,
- adding to one equation another equation multiplied by an arbitrary number.
Consider a system of linear equations:
| (1) |
We write system (1) in matrix form:
| ax=b | (2) |
  | (3) |
A is called the coefficient matrix of the system, b − right part restrictions x− vector of variables to be found. Let rank( A)=p.
Equivalent transformations do not change the rank of the coefficient matrix and the rank of the augmented matrix of the system. The set of solutions of the system also does not change under equivalent transformations. The essence of the Gauss method is to bring the matrix of coefficients A to diagonal or stepped.
Let's build the extended matrix of the system:
At the next stage, we reset all elements of column 2, below the element. If the given element is null, then this row is interchanged with the row lying below the given row and having a non-zero element in the second column. Next, we zero out all the elements of column 2 below the leading element a 22. To do this, add rows 3, ... m with row 2 multiplied by − a 32 /a 22 , ..., −a m2 / a 22, respectively. Continuing the procedure, we obtain a matrix of a diagonal or stepped form. Let the resulting augmented matrix look like:
| (7) |
 |
Because rankA=rank(A|b), then the set of solutions (7) is ( n−p) is a variety. Hence n−p unknowns can be chosen arbitrarily. The remaining unknowns from system (7) are calculated as follows. From the last equation we express x p through the rest of the variables and insert into the previous expressions. Next, from the penultimate equation, we express x p−1 through the rest of the variables and insert into the previous expressions, etc. Consider the Gauss method on specific examples.
Examples of solving a system of linear equations using the Gauss method
Example 1. Find common decision systems of linear equations by the Gauss method:
Denote by a ij elements i-th line and j-th column.
a eleven . To do this, add rows 2,3 with row 1, multiplied by -2/3, -1/2, respectively:
Matrix record type: ax=b, where
Denote by a ij elements i-th line and j-th column.
Exclude the elements of the 1st column of the matrix below the element a eleven . To do this, add rows 2,3 with row 1, multiplied by -1/5, -6/5, respectively:
We divide each row of the matrix by the corresponding leading element (if the leading element exists):
where x 3 , x
Substituting the upper expressions into the lower ones, we obtain the solution.
Then the vector solution can be represented as follows:
 |
where x 3 , x 4 are arbitrary real numbers.
Here you can solve a system of linear equations for free Gauss method online large sizes in complex numbers with a very detailed solution. Our calculator can solve online both the usual definite and indefinite system of linear equations using the Gaussian method, which has an infinite number of solutions. In this case, in the answer you will receive the dependence of some variables through others, free ones. You can also check the system of equations for compatibility online using the Gaussian solution.
About method
When solving a system of linear equations online method Gauss performs the following steps.
- We write the augmented matrix.
- In fact, the solution is divided into the forward and backward steps of the Gaussian method. The direct move of the Gauss method is called the reduction of the matrix to a stepped form. Reverse The Gaussian method is called the reduction of a matrix to a special stepped form. But in practice, it is more convenient to immediately zero out what is both above and below the element in question. Our calculator uses exactly this approach.
- It is important to note that when solving by the Gauss method, the presence in the matrix of at least one zero row with a nonzero right side(column of free members) indicates the incompatibility of the system. Solution linear system in this case does not exist.
To better understand how the Gaussian algorithm works online, enter any example, select "very detailed solution" and see its solution online.
Carl Friedrich Gauss, the greatest mathematician for a long time hesitated between philosophy and mathematics. Perhaps it was precisely such a mindset that allowed him to "leave" so noticeably in world science. In particular, by creating the "Gauss Method" ...
For almost 4 years, the articles of this site have dealt with school education, mainly from the side 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 the approach to the familiar, confusing and important areas of life gives the best results.
We humans are so arranged that no matter how much you talk about abstract thinking, but understanding always happens through examples. If there are no examples, then it is impossible to catch the principles ... How impossible it is to be on the top of a mountain otherwise than by going through its entire slope from the foot.
Same with school: for now living stories not enough we instinctively continue to regard it as a place where children are taught to understand.
For example, teaching the Gauss method...
Gauss method in the 5th grade of the school
I will make a reservation right away: the Gauss method has a much wider application, for example, when solving systems of linear equations. What we are going to talk about takes place in the 5th grade. This start, having understood which, it is much easier to understand more "advanced options". In this article we are talking about method (method) of Gauss when finding the sum of a series
Here is an example that I brought from school younger son attending the 5th grade of the Moscow gymnasium.
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 Carl (an outdated method, now not used in schools) for being,
instead of sequentially adding numbers from 1 to 100 to 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 subjected to execution in front of an astonished public. To the rest to think was disrespectful.
What did little Gauss do developed number sense? Noticed some feature number series with a constant step (arithmetic progression). AND exactly this made him later a great scientist, able to notice, possessing feeling, instinct of understanding.
This is the value of mathematics, which develops ability to see general in particular - abstract thinking. Therefore, most parents and employers instinctively consider mathematics an important discipline ...
“Mathematics should be taught later, so that it puts the mind in order.
M.V. Lomonosov".
However, the followers of those who flogged future geniuses turned the Method into something opposite. As my supervisor said 35 years ago: "They learned the question." Or, as my youngest son said yesterday about the Gauss 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 understanding of the "Queen of Sciences" by the majority.
However, let's continue...
Methods for explaining the Gauss method in the 5th grade of the school
A mathematics teacher at a Moscow gymnasium, explaining the Gauss method in Vilenkin's way, complicated the task.
What if the difference (step) of an arithmetic progression is not one, but another number? For example, 20.
The task he gave the fifth graders:
20+40+60+80+ ... +460+480+500
Before getting acquainted with the gymnasium method, let's look at the Web: how do school teachers - math tutors do it? ..
Gauss Method: Explanation #1
A well-known tutor on his YOUTUBE channel gives the following reasoning:
"let's write the numbers from 1 to 100 like this:
first a series of numbers from 1 to 50, and strictly below it another series of numbers from 50 to 100, but in 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 pass this method in the 9th grade!"
Gauss Method: Explanation #2
Another tutor, less well-known (judging by the number of views) uses more scientific approach, offering a solution algorithm of 5 points that must be performed sequentially.
For the uninitiated: 5 is one of the Fibonacci numbers traditionally considered magical. The 5-step method is always more scientific than the 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 about plus one rule : one must be added to the resulting quotient: otherwise we will get a result that is one less than the true number of pairs: 42 + 1 = 43.
This is the desired sum of the arithmetic progression from 4 to 256 with a difference of 6!
Gauss method: explanation in the 5th grade of the Moscow gymnasium
And here is how it was required to solve the problem of finding the sum of a series:
20+40+60+ ... +460+480+500
in the 5th grade of the Moscow gymnasium, Vilenkin's textbook (according to my son).
After showing the presentation, the math teacher showed a couple of Gaussian examples and gave the class the task of finding the sum of the numbers in a series with a step of 20.
This required the following:
As you can see, it is 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 followers would turn his "method" into. German teacher who flogged Karl with rods. He would have seen the symbolism and 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, do 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?
V simplification. V observation and capture simple patterns of numbers. V turning dry school arithmetic into interesting and fun activity , activating the desire to continue in the brain, and not blocking high-cost mental activity.
Is it possible to calculate the sum of the numbers of an arithmetic progression with one of the above "modifications of the Gauss method" instantly? According to the "algorithms", little Karl would have been guaranteed to avoid spanking, cultivate an aversion to mathematics and suppress his creative impulses in the bud.
Why did the tutor so insistently advise the fifth-graders "not to be afraid of misunderstanding" of the method, convincing them that they would solve "such" problems already in the 9th grade? Psychologically illiterate action. It was a good idea to note: "See? You already in the 5th grade you can solve problems that you will pass only in 4 years! What good fellows you are!"
To use the Gaussian method, level 3 of the class 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 "do not enter" how to explain the simplest things in a normal human language, not just mathematical ... They are not able to interest mathematics and completely discourage even the "capable".
Or, as my son commented, "make 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 - answers
""Look, here are the numbers from 1 to 100. What do you see?"
It's not about what the child sees. The trick is to make him 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 than he usually does"
It doesn't matter if the child sees the solution right away, it's unlikely. It is important that he ceased to be afraid to look, or as I say: "moved the task". This is the beginning of the path to understanding
"Which is easier: add, for example, 5 and 6 or 5 and 95?" A leading question... But after all, any training comes down to "guiding" a person to an "answer" - in any way acceptable to him.
At this stage, there may already be guesses about how to "save" on calculations.
All we have done is hint: the "frontal, linear" counting method is not the only one possible. If the child has truncated this, then later he will invent many more such methods, because it's interesting!!! And he will definitely avoid "misunderstanding" of mathematics, will not feel disgust for it. He got the win!
If baby discovered that adding pairs of numbers that add up to a hundred is a trifling task, then "arithmetic progression with difference 1"- a rather dreary and uninteresting thing for a child - suddenly gave life to him . Out of chaos came order, and this is always enthusiastic: that's the way we are!
A question to fill in: why, after the insight received by the child, again drive him into the framework of dry algorithms, moreover, functionally useless in this case ?!
Why make stupid rewrite sequence numbers in a notebook: so that even the capable would not have a single chance for understanding? Statistically, of course, but mass education is focused on "statistics" ...
Where did zero go?
And yet, adding up numbers that add up to 100 is much more acceptable to the mind than giving 101 ...
The "school Gauss method" requires exactly this: mindlessly fold equidistant from the center of the progression of a pair of numbers, no matter what.
What if you look?
Still, zero greatest invention humanity, 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 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.
Fuck 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 about earning bread and butter? After all, we are moving in even ranks into the era of the digital economy!
More about the school method of Gauss: "why make science out of this? .."
It was not in vain that I posted a screenshot from my son's notebook...
"What was there in the lesson?"
“Well, I immediately counted, raised my hand, but she didn’t ask. Therefore, while the others were counting, I began to do DZ 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. She said: "Wrong, you need to count as I showed!"
“It’s good that I didn’t put a deuce. And I made me write the“ course of the solution ”in their own way in a notebook. Why make a big science out of this? .."
The main crime of a math teacher
hardly after that occasion 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 have roared with indignation and, through the World Intellectual Property Organization WIPO, achieved a ban on the use of his honest name in school textbooks! ..
What main mistake school approach? Or, as I put it, the crime of school mathematics teachers against children?
Misunderstanding algorithm
What do school methodologists do, the vast majority of whom do not know how to think?
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, and not the stupidity of the school system.
What is happening: parents blame the children, and teachers ... the same for children who "do not understand mathematics! ..
Are you savvy?
What did little Carl do?
Absolutely unconventionally approached a template task. This is the quintessence of His approach. This the main thing that should be taught at 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 the Gauss method is to
what, if the number of elements in the row is odd, as in the task that was assigned to the son? ..
The "trick" is that in this case you should find the "extra" number of the series and add it to the sum of the pairs. In our example, this number is 260.
How to discover? Rewriting all pairs of numbers in a notebook!(That's why the teacher made the kids do this stupid job, trying to teach "creativity" using the Gaussian method... And that's why such a "method" is practically inapplicable to large data series, And that's why it is not a Gaussian method).
A little creativity in the school routine...
The son acted differently.
(20 + 500, 40 + 480 ...).
0+500, 20+480, 40+460 ...
Easy, 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 didn't praise, but also forced me to rewrite it "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 so that he will never forget it.
And it will step towards understanding...not just mathematics.
Admit it: how many times in your life have you added using the Gauss method? And I never!
But instinct of understanding, which develops (or extinguishes) 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 quietly entered under the strict guidance 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 operation.
Some teachers understand the absurdity of what is happening, but what to do? Law on Education, Federal State Educational Standards, methods, technological maps lessons... Everything should be done "according to and based on" and everything should be documented. Step aside - stood in line for dismissal. Let's not be hypocrites: the salary of Moscow teachers is very good... If they get fired, where should they go?..
Therefore this site not about education. He is about individual education, only possible way get out of the crowd Generation Z ...
