How to solve Cramer's method examples. Examples of solving systems of linear algebraic equations by the Cramer method

Cramer's method is based on the use of determinants in solving systems linear equations. This greatly speeds up the solution process.

Cramer's method can be used to solve a system of as many linear equations as there are unknowns in each equation. If the determinant of the system is not equal to zero, then Cramer's method can be used in the solution; if it is equal to zero, then it cannot. In addition, Cramer's method can be used to solve systems of linear equations that have a unique solution.

Definition. The determinant, composed of the coefficients of the unknowns, is called the determinant of the system and is denoted by (delta).

Determinants

are obtained by replacing the coefficients at the corresponding unknowns by free terms:

;

.

Cramer's theorem. If the determinant of the system is nonzero, then the system of linear equations has one single solution, and the unknown is equal to the ratio of the determinants. The denominator contains the determinant of the system, and the numerator contains the determinant obtained from the determinant of the system by replacing the coefficients with the unknown by free terms. This theorem holds for a system of linear equations of any order.

Example 1 Solve the system of linear equations:

According to Cramer's theorem we have:

So, the solution of system (2):

online calculator, decisive method Kramer.

Three cases in solving systems of linear equations

As appears from Cramer's theorems, when solving a system of linear equations, three cases may occur:

First case: the system of linear equations has a unique solution

(the system is consistent and definite)

Second case: the system of linear equations has countless decisions

(the system is consistent and indeterminate)

** ,

those. the coefficients of the unknowns and the free terms are proportional.

Third case: the system of linear equations has no solutions

(system inconsistent)

So the system m linear equations with n variables is called incompatible if it has no solutions, and joint if it has at least one solution. joint system equations that have only one solution is called certain, and more than one uncertain.

Examples of solving systems of linear equations by the Cramer method

Let the system

.

Based on Cramer's theorem

………….
,

where
-

system identifier. The remaining determinants are obtained by replacing the column with the coefficients of the corresponding variable (unknown) with free members:

Example 2

.

Therefore, the system is definite. To find its solution, we calculate the determinants

By Cramer's formulas we find:

So, (1; 0; -1) is the only solution to the system.

To check the solutions of systems of equations 3 X 3 and 4 X 4, you can use the online calculator, the Cramer solving method.

If there are no variables in the system of linear equations in one or more equations, then in the determinant the elements corresponding to them are equal to zero! This is the next example.

Example 3 Solve the system of linear equations by Cramer's method:

.

Decision. We find the determinant of the system:

Look carefully at the system of equations and at the determinant of the system and repeat the answer to the question in which cases one or more elements of the determinant are equal to zero. So, the determinant is not equal to zero, therefore, the system is definite. To find its solution, we calculate the determinants for the unknowns

By Cramer's formulas we find:

So, the solution of the system is (2; -1; 1).

To check the solutions of systems of equations 3 X 3 and 4 X 4, you can use the online calculator, the Cramer solving method.

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We continue to solve systems using the Cramer method together

As already mentioned, if the determinant of the system is equal to zero, and the determinants for the unknowns are not equal to zero, the system is inconsistent, that is, it has no solutions. Let's illustrate with the following example.

Example 6 Solve the system of linear equations by Cramer's method:

Decision. We find the determinant of the system:

The determinant of the system is equal to zero, therefore, the system of linear equations is either inconsistent and definite, or inconsistent, that is, it has no solutions. To clarify, we calculate the determinants for the unknowns

The determinants for the unknowns are not equal to zero, therefore, the system is inconsistent, that is, it has no solutions.

To check the solutions of systems of equations 3 X 3 and 4 X 4, you can use the online calculator, the Cramer solving method.

In problems on systems of linear equations, there are also those where, in addition to the letters denoting variables, there are also other letters. These letters stand for some number, most often a real number. In practice, such equations and systems of equations lead to search problems common properties any phenomena or objects. That is, did you invent any new material or a device, and to describe its properties, which are common regardless of the size or number of copies, it is necessary to solve a system of linear equations, where instead of some coefficients for variables there are letters. You don't have to look far for examples.

The next example is for a similar problem, only the number of equations, variables, and letters denoting some real number increases.

Example 8 Solve the system of linear equations by Cramer's method:

Decision. We find the determinant of the system:

Finding determinants for unknowns

Cramer's method or the so-called Cramer's rule is a way to search for unknown quantities from systems of equations. It can be used only if the number of required values ​​is equivalent to the number of algebraic equations in the system, that is, the main matrix formed from the system must be square and not contain zero rows, and also if its determinant must not be zero.

Theorem 1

Cramer's theorem If the main determinant $D$ of the main matrix, compiled on the basis of the coefficients of the equations, is not equal to zero, then the system of equations is consistent, and it has a unique solution. The solution of such a system is calculated using the so-called Cramer formulas for solving systems of linear equations: $x_i = \frac(D_i)(D)$

What is the Cramer method

The essence of the Cramer method is as follows:

1. To find a solution to the system by Cramer's method, first of all, we calculate the main determinant of the matrix $D$. When the calculated determinant of the main matrix, when calculated by the Cramer method, turned out to be equal to zero, then the system does not have a single solution or has an infinite number of solutions. In this case, to find a general or some basic answer for the system, it is recommended to use the Gaussian method.
2. Then you need to replace the last column of the main matrix with the column of free members and calculate the determinant $D_1$.
3. Repeat the same for all columns, getting the determinants from $D_1$ to $D_n$, where $n$ is the number of the rightmost column.
4. After all determinants of $D_1$...$D_n$ are found, the unknown variables can be calculated using the formula $x_i = \frac(D_i)(D)$.

Techniques for calculating the determinant of a matrix

To calculate the determinant of a matrix with a dimension greater than 2 by 2, several methods can be used:

• The rule of triangles, or the rule of Sarrus, resembling the same rule. The essence of the triangle method is that when calculating the determinant of the product of all numbers connected in the figure by a red line on the right, they are written with a plus sign, and all numbers connected in a similar way in the figure on the left are with a minus sign. Both rules are suitable for 3 x 3 matrices. In the case of the Sarrus rule, the matrix itself is first rewritten, and next to it, its first and second columns are rewritten again. Diagonals are drawn through the matrix and these additional columns, matrix members lying on the main diagonal or parallel to it are written with a plus sign, and elements lying on or parallel to the secondary diagonal are written with a minus sign.

Figure 1. Rule of triangles for calculating the determinant for the Cramer method

• With a method known as the Gaussian method, this method is also sometimes referred to as determinant reduction. In this case, the matrix is ​​transformed and reduced to triangular, and then multiply all the numbers on the main diagonal. It should be remembered that in such a search for a determinant, one cannot multiply or divide rows or columns by numbers without taking them out as a factor or divisor. In the case of searching for a determinant, it is only possible to subtract and add rows and columns to each other, having previously multiplied the subtracted row by a non-zero factor. Also, with each permutation of the rows or columns of the matrix, one should remember the need to change the final sign of the matrix.
• When solving Cramer's SLAE with 4 unknowns, it is best to use the Gaussian method to search and find determinants or determine the determinant through the search for minors.

Solving systems of equations by Cramer's method

We apply the Cramer method for a system of 2 equations and two required quantities:

$\begin(cases) a_1x_1 + a_2x_2 = b_1 \\ a_3x_1 + a_4x_2 = b_2 \\ \end(cases)$

Let's display it in an expanded form for convenience:

$A = \begin(array)(cc|c) a_1 & a_2 & b_1 \\ a_3 & a_4 & b_1 \\ \end(array)$

Find the determinant of the main matrix, also called the main determinant of the system:

$D = \begin(array)(|cc|) a_1 & a_2 \\ a_3 & a_4 \\ \end(array) = a_1 \cdot a_4 – a_3 \cdot a_2$

If the main determinant is not equal to zero, then to solve the slough by the Cramer method, it is necessary to calculate a couple more determinants from two matrices with the columns of the main matrix replaced by a row of free members:

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

$D_2 = \begin(array)(|cc|) a_1 & b_1 \\ a_3 & b_2 \\ \end(array) = a_1 \cdot b_2 – a_3 \cdot b_1$

Now let's find the unknowns $x_1$ and $x_2$:

$x_1 = \frac (D_1)(D)$

$x_2 = \frac (D_2)(D)$

Example 1

Cramer's method for solving a SLAE with a 3rd order (3 x 3) main matrix and three desired ones.

Solve the system of equations:

$\begin(cases) 3x_1 - 2x_2 + 4x_3 = 21 \\ 3x_1 +4x_2 + 2x_3 = 9\\ 2x_1 - x_2 - x_3 = 10 \\ \end(cases)$

We calculate the main determinant of the matrix using the above rule under paragraph number 1:

$D = \begin(array)(|ccc|) 3 & -2 & 4 \\3 & 4 & -2 \\ 2 & -1 & 1 \\ \end(array) = 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 = - 12 - 8 -12 -32 - 6 + 6 = - $64

And now three other determinants:

$D_1 = \begin(array)(|ccc|) 21 & 2 & 4 \\ 9 & 4 & 2 \\ 10 & 1 & 1 \\ \end(array) = 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 \ cdot 21 = - 84 - 40 - 36 - 160 - 18 + 42 = - $296

$D_2 = \begin(array)(|ccc|) 3 & 21 & 4 \\3 & 9 & 2 \\ 2 & 10 & 1 \\ \end(array) = 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 = - 27 + 120 + 84 – 72 + 63 – 60 = $108

$D_3 = \begin(array)(|ccc|) 3 & -2 & 21 \\ 3 & 4 & 9 \\ 2 & 1 & 10 \\ \end(array) = 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

Let's find the required values:

$x_1 = \frac(D_1) (D) = \frac(- 296)(-64) = 4 \frac(5)(8)$

$x_2 = \frac(D_1) (D) = \frac(108) (-64) = - 1 \frac (11) (16)$

$x_3 = \frac(D_1) (D) = \frac(-60) (-64) = \frac (15) (16)$

Let the system of linear equations contain as many equations as the number of independent variables, i.e. has the form

Such systems of linear equations are called quadratic. Determinant composed of coefficients at independent system variables(1.5) is called the main determinant of the system. We will denote it with the Greek letter D. Thus,

. (1.6)

If in the main determinant an arbitrary ( j th) column, replace it with the column of free members of the system (1.5), then we can get more n auxiliary determinants:

(j = 1, 2, …, n). (1.7)

Cramer's rule solving quadratic systems of linear equations is as follows. If the principal determinant D of system (1.5) is nonzero, then the system has a unique solution, which can be found by the formulas:

(1.8)

Example 1.5. Solve the system of equations using Cramer's method

.

Let us calculate the main determinant of the system:

Since D¹0, the system has a unique solution that can be found using formulas (1.8):

Thus,

Matrix Actions

1. Multiplication of a matrix by a number. The operation of multiplying a matrix by a number is defined as follows.

2. In order to multiply a matrix by a number, you need to multiply all its elements by this number. I.e

. (1.9)

Example 1.6. .

Matrix addition.

This operation is introduced only for matrices of the same order.

In order to add two matrices, it is necessary to add the corresponding elements of the other matrix to the elements of one matrix:

(1.10)
The operation of matrix addition has the properties of associativity and commutativity.

Example 1.7. .

Matrix multiplication.

If the number of matrix columns BUT matches the number of matrix rows AT, then for such matrices the operation of multiplication is introduced:

2

Thus, when multiplying the matrix BUT dimensions m´ n to matrix AT dimensions n´ k we get a matrix With dimensions m´ k. In this case, the elements of the matrix With are calculated according to the following formulas:

Problem 1.8. Find, if possible, the product of matrices AB and BA:

Decision. 1) To find a work AB, you need matrix rows A multiply by matrix columns B:

2) Artwork BA does not exist, because the number of columns of the matrix B does not match the number of matrix rows A.

Inverse matrix. Solving systems of linear equations in a matrix way

Matrix A- 1 is called the inverse of a square matrix BUT if the equality holds:

where through I denoted identity matrix the same order as the matrix BUT:

.

For a square matrix to have an inverse, it is necessary and sufficient that its determinant be nonzero. The inverse matrix is ​​found by the formula:

, (1.13)

where A ij- algebraic additions to elements aij matrices BUT(note that algebraic additions to the rows of the matrix BUT are arranged in the inverse matrix in the form of corresponding columns).

Example 1.9. Find inverse matrix A- 1 to matrix

.

We find the inverse matrix by formula (1.13), which for the case n= 3 looks like:

.

Let's find det A = | A| = 1 x 3 x 8 + 2 x 5 x 3 + 2 x 4 x 3 - 3 x 3 x 3 - 1 x 5 x 4 - 2 x 2 x 8 = 24 + 30 + 24 - 27 - 20 - 32 = - 1. Since the determinant of the original matrix is ​​different from zero, then the inverse matrix exists.

1) Find algebraic additions A ij:

For the convenience of finding inverse matrix, we placed the algebraic additions to the rows of the original matrix in the corresponding columns.

From received algebraic additions compose a new matrix and divide it by the determinant det A. Thus, we will get the inverse matrix:

Quadratic systems of linear equations with a non-zero principal determinant can be solved using an inverse matrix. For this, system (1.5) is written in matrix form:

where

Multiplying both sides of equality (1.14) on the left by A- 1 , we get the solution of the system:

, where

Thus, in order to find a solution to a square system, you need to find the inverse matrix to the main matrix of the system and multiply it on the right by the column matrix of free terms.

Problem 1.10. Solve a system of linear equations

using an inverse matrix.

Decision. We write the system in matrix form: ,

where is the main matrix of the system, is the column of unknowns, and is the column of free terms. Since the main determinant of the system , then the main matrix of the system BUT has an inverse matrix BUT-one . To find the inverse matrix BUT-1 , calculate the algebraic complements to all elements of the matrix BUT:

From the obtained numbers we compose a matrix (moreover, algebraic additions to the rows of the matrix BUT write in the appropriate columns) and divide it by the determinant D. Thus, we have found the inverse matrix:

The solution of the system is found by the formula (1.15):

Thus,

Solving Systems of Linear Equations by Ordinary Jordan Exceptions

Let an arbitrary (not necessarily square) system of linear equations be given:

(1.16)

It is required to find a solution to the system, i.e. such a set of variables that satisfies all the equalities of the system (1.16). AT general case system (1.16) can have not only one solution, but also an infinite number of solutions. It may also have no solutions at all.

In solving such problems, the well-known school course the method of elimination of unknowns, which is also called the method of ordinary Jordan eliminations. The essence of this method lies in the fact that in one of the equations of system (1.16) one of the variables is expressed in terms of other variables. Then this variable is substituted into other equations of the system. The result is a system that contains one equation and one less variable than the original system. The equation from which the variable was expressed is remembered.

This process is repeated until one last equation remains in the system. In the process of eliminating unknowns, some equations can turn into true identities, for example. Such equations are excluded from the system, since they are valid for any values ​​of the variables and, therefore, do not affect the solution of the system. If, in the process of eliminating unknowns, at least one equation becomes an equality that cannot be satisfied for any values ​​of the variables (for example, ), then we conclude that the system has no solution.

If in the course of solving inconsistent equations did not arise, then one of the remaining variables in it is found from the last equation. If only one variable remains in the last equation, then it is expressed as a number. If other variables remain in the last equation, then they are considered parameters, and the variable expressed through them will be a function of these parameters. Then the so-called reverse stroke". The found variable is substituted into the last memorized equation and the second variable is found. Then the two found variables are substituted into the penultimate memorized equation and the third variable is found, and so on, up to the first memorized equation.

As a result, we get the solution of the system. This decision will be unique if the found variables are numbers. If the first found variable, and then all the others depend on the parameters, then the system will have an infinite number of solutions (each set of parameters corresponds to a new solution). Formulas that allow finding a solution to the system depending on a particular set of parameters are called the general solution of the system.

Example 1.11.

x

After memorizing the first equation and bringing similar terms in the second and third equations, we arrive at the system:

Express y from the second equation and substitute it into the first equation:

Remember the second equation, and from the first we find z:

Making the reverse move, we successively find y and z. To do this, we first substitute into the last memorized equation , from which we find y:

.

Then we substitute and into the first memorized equation from where we find x:

Problem 1.12. Solve a system of linear equations by eliminating unknowns:

. (1.17)

Decision. Let us express the variable from the first equation x and substitute it into the second and third equations:

.

Remember the first equation

In this system, the first and second equations contradict each other. Indeed, expressing y , we get that 14 = 17. This equality is not satisfied, for any values ​​of the variables x, y, and z. Consequently, system (1.17) is inconsistent, i.e., has no solution.

Readers are invited to independently verify that the main determinant of the original system (1.17) is equal to zero.

Consider a system that differs from system (1.17) by only one free term.

Problem 1.13. Solve a system of linear equations by eliminating unknowns:

. (1.18)

Decision. As before, we express the variable from the first equation x and substitute it into the second and third equations:

.

Remember the first equation and we present similar terms in the second and third equations. We arrive at the system:

expressing y from the first equation and substituting it into the second equation , we get the identity 14 = 14, which does not affect the solution of the system, and, therefore, it can be excluded from the system.

In the last memorized equality, the variable z will be considered as a parameter. We believe . Then

Substitute y and z into the first memorized equality and find x:

.

Thus, system (1.18) has an infinite set of solutions, and any solution can be found from formulas (1.19) by choosing an arbitrary value of the parameter t:

(1.19)
Thus, the solutions of the system, for example, are the following sets of variables (1; 2; 0), (2; 26; 14), etc. Formulas (1.19) express the general (any) solution of system (1.18).

In the case when the original system (1.16) has enough a large number of equations and unknowns, the specified method of ordinary Jordanian eliminations seems cumbersome. However, it is not. It is enough to derive an algorithm for recalculating the coefficients of the system at one step in general view and formalize the solution of the problem in the form of special Jordan tables.

Let a system of linear forms (equations) be given:

, (1.20)
where xj- independent (desired) variables, aij- constant coefficients
(i = 1, 2,…, m; j = 1, 2,…, n). Right parts of the system y i (i = 1, 2,…, m) can be both variables (dependent) and constants. It is required to find solutions to this system by eliminating unknowns.

Let us consider the following operation, hereinafter referred to as "one step of ordinary Jordan exceptions". From an arbitrary ( r th) equality, we express an arbitrary variable ( x s) and substitute into all other equalities. Of course, this is only possible if a rs¹ 0. Coefficient a rs is called the resolving (sometimes guiding or main) element.

We'll get next system:

. (1.21)

From s th equality of system (1.21), we will subsequently find the variable x s(after other variables are found). S The th line is remembered and subsequently excluded from the system. The remaining system will contain one equation and one less independent variable than the original system.

Let us calculate the coefficients of the resulting system (1.21) in terms of the coefficients of the original system (1.20). Let's start with r th equation, which, after expressing the variable x s through the rest of the variables will look like this:

Thus, the new coefficients r th equation are calculated by the following formulas:

(1.23)
Let us now calculate the new coefficients b ij(i¹ r) arbitrary equation. To do this, we substitute the variable expressed in (1.22) x s in i th equation of system (1.20):

After bringing like terms, we get:

(1.24)
From equality (1.24) we obtain formulas by which the remaining coefficients of system (1.21) are calculated (with the exception of r th equation):

(1.25)
The transformation of systems of linear equations by the method of ordinary Jordanian eliminations is presented in the form of tables (matrices). These tables are called "Jordan tables".

Thus, problem (1.20) is associated with the following Jordan table:

Table 1.1

	x 1	x 2	…	xj	…	x s	…	x n
y 1 =	a 11	a 12		a 1j		a 1s		a 1n
…………………………………………………………………..
y i=	a i 1	a i 2		aij		a is		a in
…………………………………………………………………..
y r=	a r 1	a r 2		a rj		a rs		a rn
………………………………………………………………….
y n=	a m 1	a m 2		a mj		a ms		amn

Jordan table 1.1 contains the left head column, in which the right parts of the system (1.20) are written, and the top head line, in which the independent variables are written.

The remaining elements of the table form the main matrix of coefficients of system (1.20). If we multiply the matrix BUT to the matrix consisting of the elements of the upper header row, then we get the matrix consisting of the elements of the left header column. That is, in essence, the Jordan table is a matrix form of writing a system of linear equations: . In this case, the following Jordan table corresponds to system (1.21):

Table 1.2

	x 1	x 2	…	xj	…	y r	…	x n
y 1 =	b 11	b 12		b 1 j		b 1 s		b 1 n
…………………………………………………………………..
y i =	b i 1	b i 2		b ij		b is		b in
…………………………………………………………………..
x s =	br 1	br 2		b rj		brs		b rn
………………………………………………………………….
y n =	b m 1	b m 2		bmj		b ms		bmn

Permissive element a rs we will highlight in bold. Recall that in order to implement one step of Jordan exceptions, the resolving element must be nonzero. A table row containing a permissive element is called a permissive row. The column containing the enable element is called the enable column. When moving from a given table to the next table, one variable ( x s) from the top header row of the table is moved to the left header column and, conversely, one of the free members of the system ( y r) is moved from the left header column of the table to the top header row.

Let us describe the algorithm for recalculating the coefficients in passing from the Jordan table (1.1) to the table (1.2), which follows from formulas (1.23) and (1.25).

1. The enabling element is replaced by the inverse number:

2. The remaining elements of the permissive line are divided by the permissive element and change sign to the opposite:

3. The remaining elements of the enabling column are divided into the enabling element:

4. Elements that are not included in the resolving row and resolving column are recalculated according to the formulas:

The last formula is easy to remember if you notice that the elements that make up the fraction , are at the intersection i-oh and r-th lines and j th and s-th columns (resolving row, resolving column and the row and column at the intersection of which the element to be recalculated is located). More precisely, when memorizing the formula you can use the following chart:

-21 -26 -13 -37

Performing the first step of the Jordanian exceptions, any element of Table 1.3 located in the columns x 1 ,…, x 5 (all specified elements are not equal to zero). You should not only select the enabling element in the last column, because need to find independent variables x 1 ,…, x 5 . We choose, for example, the coefficient 1 with a variable x 3 in the third row of table 1.3 (the enabling element is shown in bold). When moving to table 1.4, the variable x The 3 from the top header row is swapped with the constant 0 of the left header column (third row). At the same time, the variable x 3 is expressed in terms of the remaining variables.

string x 3 (Table 1.4) can, having previously remembered, be excluded from Table 1.4. Table 1.4 also excludes the third column with a zero in the upper header line. The point is that regardless of the coefficients of this column b i 3 all terms corresponding to it of each equation 0 b i 3 systems will be equal to zero. Therefore, these coefficients can not be calculated. Eliminating one variable x 3 and remembering one of the equations, we arrive at a system corresponding to Table 1.4 (with the line crossed out x 3). Choosing in table 1.4 as a resolving element b 14 = -5, go to table 1.5. In table 1.5, we remember the first row and exclude it from the table along with the fourth column (with zero at the top).

Table 1.5 Table 1.6

From last table 1.7 we find: x 1 = - 3 + 2x 5 .

Sequentially substituting the already found variables into the memorized lines, we find the remaining variables:

Thus, the system has an infinite number of solutions. variable x 5 , you can assign arbitrary values. This variable acts as a parameter x 5 = t. We proved the compatibility of the system and found it common decision:

x 1 = - 3 + 2t

x 2 = - 1 - 3t

x 3 = - 2 + 4t . (1.27)
x 4 = 4 + 5t

x 5 = t

Giving parameter t various meanings, we get an infinite number of solutions to the original system. So, for example, the solution of the system is the following set of variables (- 3; - 1; - 2; 4; 0).

In the first part, we looked at some theoretical material, the substitution method, and the method of term-by-term addition of system equations. To everyone who came to the site through this page, I recommend that you read the first part. Perhaps some visitors will find the material too simple, but in the course of solving systems of linear equations, I made a series of very important notes and conclusions regarding the decision math problems generally.

And now we will analyze Cramer's rule, as well as the solution of a system of linear equations using the inverse matrix (matrix method). All materials are presented simply, in detail and clearly, almost all readers will be able to learn how to solve systems using the above methods.

We first consider Cramer's rule in detail for a system of two linear equations in two unknowns. What for? - After all the simplest system can be solved by the school method, by term addition!

The fact is that even if sometimes, but there is such a task - to solve a system of two linear equations with two unknowns using Cramer's formulas. Secondly, a simpler example will help you understand how to use Cramer's rule to more difficult case– systems of three equations with three unknowns.

In addition, there are systems of linear equations with two variables, which it is advisable to solve exactly according to Cramer's rule!

Consider the system of equations

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

Gauss method.

If , then the system has a unique solution, and to find the roots, we must calculate two more determinants:
and

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

The roots of the equation are found by the formulas:
,

Example 7

Solve a system of linear equations

Decision: We see that the coefficients of the equation are quite large, on the right side there are decimals with a comma. The comma is a 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 in terms of another, but in this case, you will surely get terrible fancy fractions, which are extremely inconvenient to work with, and the design of the solution will look just awful. You can multiply the second equation by 6 and subtract term by term, but the same fractions will appear here.

What to do? In such cases, Cramer's formulas come to the rescue.

;

;

Answer: ,

Both roots have infinite tails and are found approximately, which is quite acceptable (and even commonplace) for econometrics problems.

Comments are not needed here, since the task is solved according to ready-made formulas, however, there is one caveat. When use this method, compulsory The fragment of the assignment is the following fragment: "so the system has a unique solution". Otherwise, the reviewer may punish you for disrespecting Cramer's theorem.

It will not be superfluous to check, which is convenient to carry out on a calculator: we substitute the approximate values ​​\u200b\u200bin the left side of each equation of the system. As a result, with a small error, numbers that are on the right side should be obtained.

Example 8

Express your answer in ordinary improper fractions. Make a check.

This is an example for independent decision(example of finishing and answer at the end of the lesson).

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

We find the main determinant of the system:

If , then the system has infinitely many solutions or is inconsistent (has no solutions). In this case, Cramer's rule will not help, you need to use the Gauss method.

If , then the system has a unique solution, and to find the roots, we must calculate three more determinants:
, ,

And finally, the answer is calculated by the formulas:

As you can see, the “three by three” case is fundamentally no different from the “two by two” case, the column of free terms sequentially “walks” from left to right along the columns of the main determinant.

Example 9

Solve the system using Cramer's formulas.

Decision: Let's solve the system using Cramer's formulas.

, so the system has a unique solution.

Answer: .

Actually, there is nothing special to comment here again, in view of the fact that the decision is made according to ready-made formulas. But there are a couple of notes.

It happens that as a result of calculations, “bad” irreducible fractions are obtained, for example: .
I recommend the following "treatment" algorithm. If there is no computer at hand, we do this:

1) There may be a mistake in the calculations. As soon as you encounter a “bad” shot, you must immediately check whether is the condition rewritten correctly. If the condition is rewritten without errors, then you need to recalculate the determinants using the expansion in another row (column).

2) If no errors were found as a result of the check, then most likely a typo was made in the condition of the assignment. In this case, calmly and CAREFULLY solve the task to the end, and then make sure to check and draw it up on a clean copy after the decision. Of course, checking a fractional answer is an unpleasant task, but it will be a disarming argument for the teacher, who, well, really likes to put a minus for any bad thing like. How to deal with fractions is detailed in the answer for Example 8.

If you have a computer at hand, then use an automated program to check it, which can be downloaded for free at the very beginning of the lesson. By the way, it is most advantageous to use the program right away (even before starting the solution), you will immediately see the intermediate step at which you made a mistake! The same calculator automatically calculates the solution of the system matrix method.

Second remark. From time to time there are systems in the equations of which some variables are missing, for example:

Here in the first equation there is no variable , in the second there is no variable . In such cases, it is very important to correctly and CAREFULLY write down the main determinant:
– zeros are put in place of missing variables.
By the way, it is rational to open determinants with zeros in the row (column) in which zero is located, since there are noticeably fewer calculations.

Example 10

Solve the system using Cramer's formulas.

This is an example for self-solving (finishing sample and answer at the end of the lesson).

For the case of a system of 4 equations with 4 unknowns, Cramer's formulas are written according to similar principles. You can see a live example in the Determinant Properties lesson. Reducing the order of the determinant - five 4th order determinants are quite solvable. Although the task is already very reminiscent of a professor's shoe on the chest of a lucky student.

Solution of the system using the inverse matrix

The inverse matrix method is essentially special case matrix equation(See Example No. 3 of the specified lesson).

To study this section, you need to be able to expand the determinants, find the inverse matrix and perform matrix multiplication. Relevant links will be given as the explanation progresses.

Example 11

Solve the system with the matrix method

Decision: We write the system in matrix form:
, where

Please look at the system of equations and the matrices. By what principle we write elements into matrices, I think everyone understands. The only comment: if some variables were missing in the equations, then zeros would have to be put in the corresponding places in the matrix.

We find the inverse matrix by the formula:
, where is the transposed matrix of algebraic complements of the corresponding elements of the matrix .

First, let's deal with the determinant:

Here the determinant is expanded by the first line.

Attention! If , then the inverse 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 elimination of unknowns (Gauss method).

Now you need to calculate 9 minors and write them into the matrix of minors

Reference: It is useful to know the meaning of double subscripts in linear algebra. The first digit is the line number in which the element is located. The second digit is the number of the column in which the element is located:

That is, a double subscript indicates that the element is in the first row, third column, while, for example, the element is in the 3rd row, 2nd column

Consider a system of 3 equations with three unknowns

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

(2.4)

if 0. Here

It is Cramer's rule solving a system of three linear equations in three unknowns.

Example 2.3. Solve a system of linear equations using Cramer's rule:

Decision . Finding the determinant of the main matrix of the system

Since 0, then to find a solution to the system, you can apply Cramer's rule, but first calculate three more determinants:

Examination:

Therefore, the solution is found correctly. 

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

Cramer's theorem. Quadratic system of linear equations with a non-zero determinant of the main matrix of the system (0) has one and only one solution, and this solution is calculated by the formulas

(2.5)

where  – main matrix determinant,  i – matrix determinant, derived from the main, replacementith column free members column.

Note that if =0, then Cramer's rule is not applicable. This means that the system either has no solutions at all, or has infinitely many solutions.

Having formulated Cramer's theorem, the question naturally arises of calculating higher-order determinants.

2.4. nth order determinants

Additional minor M ij element a ij is called the determinant obtained from the given by deleting i-th line and j-th column. Algebraic addition A ij element a ij is called the minor of this element, taken with the sign (–1) i + j, i.e. A ij = (–1) i + j M ij .

For example, let's find minors and algebraic complements of elements a 23 and a 31 determinants

We get



Using the concept of algebraic complement, we can formulate the determinant expansion theoremn-th order by row or column.

Theorem 2.1. Matrix determinantAis equal to the sum of the products of all elements of some row (or column) and their algebraic complements:

(2.6)

This theorem underlies one of the main methods for calculating determinants, the so-called. order reduction method. As a result of the expansion of the determinant n th order in any row or column, we get n determinants ( n–1)-th order. In order to have fewer such determinants, it is advisable to choose the row or column that has the most zeros. In practice, the expansion formula for the determinant is usually written as:

those. algebraic additions are written explicitly in terms of minors.

Examples 2.4. Calculate the determinants by first expanding them in any row or column. Usually in such cases, choose the column or row that has the most zeros. The selected row or column will be marked with an arrow.



2.5. Basic properties of determinants

Expanding the determinant in any row or column, we get n determinants ( n–1)-th order. Then each of these determinants ( n–1)-th order can also be decomposed into a sum of determinants ( n–2)th order. Continuing this process, one can reach the determinants of the 1st order, i.e. to the elements of the matrix whose determinant is being calculated. So, to calculate the 2nd order determinants, you will have to calculate the sum of two terms, for the 3rd order determinants - the sum of 6 terms, for the 4th order determinants - 24 terms. The number of terms will increase sharply as the order of the determinant increases. This means that the calculation of determinants of very high orders becomes a rather laborious task, beyond the power of even a computer. However, determinants can be calculated in another way, using the properties of determinants.

Property 1 . The determinant will not change if rows and columns are swapped in it, i.e. when transposing a matrix:

.

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

Property 2 . The determinant changes sign when two rows (columns) are interchanged.

Consequence . If the determinant has two identical rows (columns), then it is equal to zero.

Property 3 . The common factor of all elements in any row (column) can be taken out of the sign of the determinant.

For example,

Consequence . If all elements of some row (column) of the determinant are equal to zero, then the determinant itself is equal to zero.

Property 4 . The determinant will not change if the elements of one row (column) are added to the elements of another row (column) multiplied by some number.

For example,

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