How to solve the Equation of the Cramer formula examples. Linear equations

In order to master this paragraph, you should be able to disclose the identifiers "two two" and "three to three". If the determinants are bad, please study the lesson How to calculate the determinant?

First, we will consider in detail the ruler of the crater for the system of two linear equations With two unknowns. What for? - After all simpler system You can solve the school method, the method of killing addition!

The fact is that even if it is sometimes, this task is found - to solve the system of two linear equations with two unknowns by crawler formulas. Secondly, a simpler example will help to understand how to use the crawler rule for a more complex case - the systems of three equations with three unknowns.

In addition, there are systems of linear equations with two variables that it is advisable to solve precisely according to the rule of Cramer!

Consider the system of equations

In the first step, we calculate the determinant, it is called the main determinant of the system.

Gauss method.

If, the system has a single decision, and for finding the roots, we must calculate two more determinants:
and

In practice, the above determinants may also be denoted by the Latin letter.

The roots of the equations are found by formulas:
,

Example 7.

Solve the system of linear equations

Decision: We see that the coefficients of the equation are large enough, there are present in the right decimal fractions With comma. Comma - rather rare guest in practical tasks In mathematics, I took this system from an econometric problem.

How to solve such a system? You can try to express one variable across the other, but in this case it will certainly get terrible trousers, with which it is extremely inconvenient to work, and the decoration of the solution will look just awful. You can multiply the second equation on 6 and carry out the soil subtraction, but also the same fractions will arise.

What to do? In such cases, they come to the aid of the formula of the crater.

;

;

Answer: ,

Both roots have endless tails, and are found approximately, which is quite acceptable (and even ordinary) for the problems of econometrics.

Comments are not needed here, since the task is solved on the finished formulas, however, there is one nuance. When using this method, compulsorythe task design fragment is the following fragment: "So the system has a single decision". Otherwise, the reviewer may punish you for disrespecting the Cramer Theorem.

At all, it will not be superfluous, which is convenient to carry out on the calculator: we substitute approximate values \u200b\u200binto the left part of each equation of the system. As a result, with a small error, the numbers that are in the right parts should be turned out.

Example 8.

Answer to submit to ordinary irregular fractions. Make check.

This is an example for self-decide (An example of a clean design and response at the end of the lesson).

We turn to the consideration of the Cramer rule for a system of three equations with three unknowns:

We find the main determinant of the system:

If, the system has infinitely many solutions or inconspicuous (not solutions). In this case, the Rule of Cramer will not help, you need to use the Gauss method.

If, the system has a single solution and for finding the roots, we must calculate three more determinants:
, ,

And finally, the answer is calculated by the formulas:

As you can see, the case of "three to three" does not differ in principle from the case of "two two", the column of free members consistently "stroll" from left to right through the columns of the main determinant.

Example 9.

Solve the system according to the crawler formulas.

Decision: Resolving the system according to the crawler formulas.

So the system has a single solution.

Answer: .

Actually, there is nothing more to comment here again, in view of the fact that the decision passes through the finished formulas. But there is a couple of comments.

It happens that as a result of calculations, "bad" non-interpretable fractions are obtained, for example:.
I recommend the next treatment algorithm. If there is no computer at hand, do this:

1) An error in calculations is allowed. As soon as you encountered a "bad" fraction, immediately need to check, conductive conditioner correctly. If the condition is rewritten without errors, then you need to recalculate the determinants using the decomposition on another line (column).

2) If the error check is not detected, it is likely to be a typo in the assignment condition. In this case, calmly and carefully turn the task to the end, and then be sure to check And we make it on the finishing after the decision. Of course, the verification of a fractional response is an unpleasant, but it will be a disarming argument for a teacher who really loves to put minus for any bjaka like. How to manage with fractions, detailed in response for example 8.

If there is a computer at hand, then use the automated program to be downloaded for free at the very beginning of the lesson. By the way, it is most advantageous to immediately use the program (even before the decision), you will immediately see the intermediate step on which the error was allowed! The same calculator automatically calculates the system solution. matrix method.

Remark Second. From time to time there are systems in the equations of which there are no variables, for example:

Here in the first equation there is no variable, in the second - variable. In such cases, it is very important to correctly and carefully record the main identifier:
- On the site of the missing variables are zeros.
By the way, the determinants with zeros are rationally disclosed along the line (column), which is zero, since the calculations are noticeably less.

Example 10.

Solve the system according to the crawler formulas.

This is an example for an independent solution (a sample of clean design and response at the end of the lesson).

For the case of a system of 4 of equations with 4 unknown, the Cramer formula is recorded by similar principles. A living example can be viewed at the lesson properties of the determinant. A decrease in the order of the determinant - the five determinants of the 4th order are completely solid. Although the task is already quite reminded by the professor's boot on the chest at the lucky student.

Solution of the system with a return matrix

Method reverse matrix - this is essentially a special case matrix equation (See Example number 3 of the specified lesson).

To explore this section, you must be able to disclose the determinants, find a reverse matrix and perform matrix multiplication. Relevant links will be given in the course of the explanation.

Example 11.

Solve the system with a matrix method

Decision: Write the system in matrix form:
where

Please look at the system of equations and matrix. According to which principle, write elements in the matrix, I think everyone is understandable. The only comment: if there were no variables in the equations, then at the appropriate places in the matrix it would be necessary to put zeros.

Reverse matrix we find by the formula:
where - a transposed matrix of algebraic additions to the corresponding elements of the matrix.

First we deal with the determinant:

Here the determinant is disclosed on the first line.

Attention! If, then the return matrix does not exist, and it is impossible to solve the system by the matrix method. In this case, the system is solved by the exclusion of unknown (Gauss method).

Now you need to calculate 9 minors and record them in the Mind Matrix

Reference: It is useful to know the meaning of dual substitution indices in a linear algebra. The first digit is the line number in which this item is located. The second digit is the column number in which this item is:

That is, a double substitution index indicates that the element is in the first row, the third column, and, for example, the element is in 3 string, 2 columns

During the solution, the calculation of minor reinstall is better to paint in detail, although, with a certain experience, they can be adopted to read with errors orally.

Methods Cramer and Gaussa - Some of the most popular solutions Slough. In addition, in some cases it is advisable to use specific methods. The session is close, and now it's time to repeat or master them from scratch. Today we deal with the solution by the Cramer method. After all, the solution of the system of linear equations by the Cramer method is a very useful skill.

Linear algebraic equations systems

Linear system algebraic equations - System of equations of the form:

Set of values x. in which the system equations appeal in identities, is called the solution solving, a. and b. - Real coefficients. A simple system consisting of two equations with two unknowns can be solved in the mind either expressing one variable through the other. But variables (ICs) in the Slava can be much more than two, and here it is not enough for simple school manipulations. What to do? For example, to solve the Slava method of Kramer!

So let the system consist of n. Equations S. n. unknown.

Such a system can be rewritten in matrix form

Here A. - the main matrix of the system, X. and B. , accordingly, the matrices-columns of unknown variables and free members.

Solution of Slava Method of Cramer

If the determinant of the main matrix is \u200b\u200bnot equal to zero. (The matrix is \u200b\u200bnondegenerate), the system can be solved according to the Cramer method.

According to the Cramer method, the solution is by formulas:

Here delta - determinant of the main matrix, and delta X. N-NO - the determinant obtained from the determinant of the main matrix by replacing the N-number column on the column of free members.

This is the entire essence of the Cramer method. Substituting the values \u200b\u200bfound according to the above formulas x. In the desired system, we are convinced of the correctness (or vice versa) of our decision. So that you caught the essence faster, give below detailed solution Slava method of cramer:

Even if you do not work from the first time, do not worry! A little practice, and you will start clicking the Slava as nuts. Moreover, now it is absolutely optionally to sober over the notebook, solving bulky calculations and by writing the rod. You can easily solve the sample by the drive method online, only substituting in finished shape coefficients. Try online calculator The solutions by the Cramer can, for example, on this site.

And if the system turned out to be stubborn and not surrender, you can always seek help to our authors, for example, to. Be in the system at least 100 unknown, we will definitely solve it true and exactly on time!

With the number of equations the same with the number of unknowns with the main determinant, the matrix, which is not zero, system coefficients (for such equations, the solution is only one).

Cramer Theorem.

When the determinant of the matrix of the square system is nonzero, it means that the system is in order to have one solution and it can be found on cramer formulas:

where Δ - system matrix determinant,

Δ I. - determinant of the system matrix, in which instead i.The column is the column of the right parts.

When the determinant of the system is zero, it means that the system can become joint or inconception.

This method is usually used for small systems with volume computing and if when it is necessary to determine 1-well from unknown. The complexity of the method is that it is necessary to calculate a lot of determinants.

Description of the Cramer method.

There is a system of equations:

The system of 3 equations can be solved by the Cramer method, which is discussed above for a system of 2 equations.

We make a determinant of the coefficients of unknown:

It will be system Determined. When D ≠ 0So, the system is coordinated. Now make up 3 additional identifiers:

,,

We solve the system of PO cramer formulas:

Examples of solving systems of equations by the Cramer method.

Example 1..

Dana System:

By solving it by the Cramer method.

First you need to calculate the determinant of the system matrix:

Because Δ ≠ 0, it means, from the Cramer Theorem, the system is co-developed and it has one solution. Calculate additional identifiers. The determinant δ 1 is obtained from the determinant δ, replacing its first column by a column of free coefficients. We get:

In the same way, we obtain the determinant δ 2 from the system matrix determinant replacing the second column by a column of free coefficients:

Cramer method or so-called crawler rule is a way to search for unknown values \u200b\u200bfrom equations systems. It can only be used if the number of desired values \u200b\u200bis equivalent to the number of algebraic equations in the system, that is, the main matrix formed from the system must be square and do not contain zero lines, and also if its determinant should not be zero.

Theorem 1.

Kramera theorem If the main determinant of the $ d $ of the main matrix, composed on the basis of the coefficients of equations, is not zero, then the system of equations is coordinated, and the solution has the only one. The solution of such a system is calculated through the so-called Cramer formulas to solve linear equations: $ x_i \u003d \\ FRAC (D_I) (D) $

What is the Cramer method

The essence of the Cramer method is as follows:

1. To find the solution of the system by the Cramer method, the first thing is calculated the main determinant of the $ D $ matrix. When the calculated determinant of the main matrix, when calculating the Cramer method turned out to be zero, then the system does not have a single solution or has an endless amount of solutions. In this case, it is recommended to apply the Gauss method to find a common or basic response for the system.
2. Then you need to replace the extreme column of the main matrix on the column of free members and calculate the identifier $ d_1 $.
3. Repeat the same for all columns, having received the determinants from $ d_1 $ to $ d_n $, where $ n $ is the number of the extreme right column.
4. After all the determinants are found $ d_1 $ ... $ d_n $, you can calculate unknown variables by $ x_i \u003d \\ frac formula (d_i) (D) $.

Receptions for calculating the determinant of the matrix

To calculate the determinant of the matrix with a dimension of more than 2 to 2, you can use several ways:

• The rule of triangles, or the Sarrusus rule, resembling the same rule. The essence of the triangle method is that when calculating the determinant, the product of all numbers connected in the figure of the red line on the right is recorded with a plus sign, and all the numbers connected in the same way in the figure on the left - with a minus sign. B, then, and another rule is suitable for matrices of 3 x 3. 3. In the case of the Sarruska rule, the matrix itself corresponds first, and next to it is rewritten to its first and second column. Through the matrix and these additional columns are diagonally, the matrix members lying on the main diagonal or on the parallel to it are recorded with the plus sign, and the elements lying on the side diagonal or in parallel to it - with a minus sign.

Figure 1. Triangle rule to calculate the determinant for the Cramer method

• Using the method known as the Gauss method, also sometimes this method is called a decrease in the order of the determinant. In this case, the matrix is \u200b\u200bconverted and driven to a triangular form, and then multiplious all the numbers standing on the main diagonal. It should be remembered that with such a search for the determinant, it is impossible to multiply or divide the lines or columns into numbers without making them as a multiplier or divider. In the case of a search for a determinant, it is possible to only deduct and fold the strings and pillars between themselves, after pre-mowing the subtracted line to the nonzero multiplier. Also, with each permutation of the lines or columns of the matrix, the places should be remembered about the need to change the final sign in the matrix.
• When solving the Cramer method, the Slava with 4 unknowns is best to use exactly the Gauss method for searching and finding identifiers or define the determinant through the search for minorors.

Solution of systems of equations by Cramer

Applicable Cramer Method for a system of 2 equations and two desired values:

$ \\ begin (Cases) a_1x_1 + a_2x_2 \u003d b_1 \\\\ a_3x_1 + a_4x_2 \u003d b_2 \\ End (Cases) $

Display it in an extended form for convenience:

$ A \u003d \\ begin (array) (CC | C) A_1 & A_2 & B_1 \\\\ A_3 & A_4 & B_1 \\ END (Array) $

We will find the determinant of the main matrix, also called the main determinant of the system:

$ D \u003d \\ begin (array) (| CC |) A_1 & A_2 \\\\ A_3 & A_4 \\ END (Array) \u003d A_1 \\ CDOT A_4 - A_3 \\ CDOT A_2 $

If the main determinant is not equal to zero, it is necessary to calculate a couple of determinants from two matrices with a replaced column of the main matrix on the line of free members to solve the sample method:

$ D_1 \u003d \\ begin (array) (| cc |) b_1 & a_2 \\\\ b_2 & a_4 \\\\ \\ end (array) \u003d b_1 \\ cdot a_4 - b_2 \\ cdot a_4 $

$ D_2 \u003d \\ begin (array) (| CC |) A_1 & B_1 \\\\ A_3 & B_2 \\\\ \\ END (Array) \u003d A_1 \\ CDOT B_2 - A_3 \\ CDOT B_1 $

Now find unknown $ x_1 $ and $ x_2 $:

$ x_1 \u003d \\ FRAC (D_1) (D) $

$ x_2 \u003d \\ FRAC (D_2) (D) $

Example 1.

Cramer Method for solving a slope with the main matrix 3 of the order (3 x 3) and three is the desired.

Solve the system of equations:

$ \\ begin (Cases) 3x_1 - 2x_2 + 4x_3 \u003d 21 \\\\ 3x_1 + 4x_2 + 2x_3 \u003d 9 \\\\ 2X_1 - X_2 - X_3 \u003d 10 \\ END (Cases) $

Consider the chief determinant of the matrix using the above-mentioned number 1 rule:

$ D \u003d \\ begin (array) (| CCC |) 3 & -2 & 4 \\\\ 3 & 4 & -2 \\ ED (Array) \u003d 3 \\ CDOT 4 \\ CDOT ( -1) + 2 \\ Cdot (-2) \\ CDOT 2 + 4 \\ CDOT 3 \\ CDOT (-1) - 4 \\ CDOT 4 \\ CDOT 2 - 3 \\ CDOT (-2) \\ CDOT (-1) - (- 1) \\ CDOT 2 \\ CDOT 3 \u003d - 12 - 8 -12 -32 - 6 + 6 \u003d - 64 $

And now three other determinants:

$ D_1 \u003d \\ begin (array) (| CCC |) 21 & 2 & 4 \\\\ 9 & 4 & 2 \\\\ 10 & 1 & 1 \\ END (Array) \u003d 21 \\ CDOT 4 \\ CDOT 1 + (- 2) \\ CDOT 2 \\ CDOT 10 + 9 \\ CDOT (-1) \\ CDOT 4 - 4 \\ CDOT 4 \\ CDOT 10 - 9 \\ CDOT (-2) \\ CDOT (-1) - (-1) \\ CDOT 2 \\ $ D_2 \u003d \\ begin (array) (| CCC |) 3 & 21 & 4 \\\\ 3 & 9 & 2 \\ ED (Array) \u003d 3 \\ CDOT 9 \\ CDOT (- 1) + 3 \\ CDOT 10 \\ CDOT 4 + 21 \\ CDOT 2 \\ CDOT 2 - 4 \\ CDOT 9 \\ CDOT 2 - 21 \\ CDOT 3 \\ CDOT (-1) - 2 \\ CDOT 10 \\ CDOT 3 \u003d - 27 + 120 + 84 - 72 + 63 - 60 \u003d 108 $

$ D_3 \u003d \\ begin (array) (| CCC |) 3 & -2 & 21 \\\\ 3 & 4 & 9 \\\\ 2 & 1 & 10 \\ END (Array) \u003d 3 \\ CDOT 4 \\ CDOT 10 + 3 \\ Cdot (-1) \\ CDOT 21 + (-2) \\ CDOT 9 \\ CDOT 2 - 21 \\ CDOT 4 \\ CDOT 2 - (-2) \\ CDOT 3 \\ CDOT 10 - (-1) \\ CDOT 9 \\ CDOT 3 \u003d 120 - 63 - 36 - 168 + 60 + 27 \u003d - $ 60

Find the desired values:

$ X_1 \u003d \\ FRAC (D_1) (D) \u003d \\ FRAC (- 296) (- 64) \u003d 4 \\ FRAC (5) (8) $

$ X_2 \u003d \\ FRAC (D_1) (D) \u003d \\ FRAC (108) (-64) \u003d - 1 \\ FRAC (11) (16) $

$ X_3 \u003d \\ FRAC (D_1) (D) \u003d \\ FRAC (-60) (-64) \u003d \\ FRAC (15) (16) $

Consider a system of 3 equations with three unknown

Using the determinants of the 3rd order, the solution of such a system can be written in the same form as for the system of two equations, i.e.

If 0. Here

(2.4)

it is

Kramer rule solutions of a system of three linear equations with three unknown example 2.3..

Solve a system of linear equations using the crawler rule: . We find the determinant of the main system matrix

Decision Since 0, to find a system solution, you can apply the craver rule, but one more than three more determinants are pre-calculated:

Check:

Consequently, the solution is found correctly. 

Cramer rules obtained for

linear systems the 2nd and 3rd Order, suggest that the same rules can be formulated for both linear systems of any order. Really takes place Cramer Theorem.

Square system of linear equations with different from zero determinant of the main system matrix It has one and only one solution and this decision is calculated by the formulas. (0) where

(2.5)

the determinant of the main matrix  – the determinant of the matrix,  i. – obtained from the main, replacement, -to column column of free membersi..

Note that if  \u003d 0, then the craver rule is not applicable. This means that the system either does not have any solutions, or has infinitely many solutions.

Formulating the theorem of the Cramer, the question of the calculation of the highest order determinants naturally arises.

2.4. N-order determinants

Additional minor M. iJ. Element a. iJ. called the determinant derived from this by crossing i.- Row I. j.-to column. Algebraic supplement A. iJ. Element a. iJ. called minor of this element taken with a sign (-1) i. + j. . A. iJ. = (–1) i. + j. M. iJ. .

For example, we find minors and algebraic elements supplements a. 23 I. a. 31 determinants

Receive



Using the concept of algebraic supplement can be formulated theorem on the decomposition of the determinantn.-o order row or column.

Theorem 2.1. The determinant of the matrixA. equal to the amount of works of all elements of a certain line (or column) on their algebraic additions:

(2.6)

This theorem underlies one of the main methods for calculating the determinants, the so-called. method lowering order. As a result of the decomposition of the determinant n.-o order for any row or column, it obtains N determinants ( n.-1) -go order. So that such determinants were less, it is advisable to choose the line or column in which the most zeros. In practice, the definition formula of the determinant is usually written in the form:

those. Algebraic additions are recorded explicitly through the minors.

Examples 2.4. Calculate the determinants, pre-laying them on any row or column. Usually in such cases choose such a column or a string in which the most zeros. The selected string or column will be denoted by the arrow.



2.5. The main properties of determinants

Decomposing the determinant for any string or column, we get N determinants ( n.-1) -go order. Then each of these determinants ( n.-1) -go order can also be decomposed in the amount of determinants ( n.-2) -go order. Continuing this process, you can walk to the determinants of the 1st order, i.e. Before the elements of the matrix, the determinant of which is calculated. So, to calculate the determinants of the 2nd order will have to calculate the sum of the two terms, for the determinants of the 3rd order - the amount of the 6 components, for the 4th order determinants - 24 terms. The number of components will sharply increase as the order of the determinant increases. This means that the calculation of identifiers of very high orders becomes a rather laborious task, unbearable even for computer. However, the determinants can be calculated differently using the properties of the determinants.

Property 1. . The determinant will not change if it is swapped in places and columns, i.e. When transposing the matrix:

.

This property indicates equality of the rows and columns of the determinant. In other words, any statement about the columns of the determinant is true for its rows and vice versa.

Property 2. . The determinant changes the sign when rearranging two lines (columns).

Corollary . If the determinant has two identical strings (column), then it is zero.

Property 3. . The total multiplier of all elements in any row (column) can be reached by the identifier sign..

For example,

Corollary . If all the elements of a certain line (column) of the determinant are zero, then the determinant itself is zero.

Property 4. . The determinant will not change if to the elements of one line (column), add the elements of another string (column) multiplied by any number.

For example,

Property 5. . The determinant of the works of matrices is equal to the product of determinants of the matrices: