Algorithm for solving linear systems of third order differential equations. Cramer's method for solving systems of linear equations Solving systems of linear equations"
Matrices. Actions on matrices. Properties of operations on matrices. Types of matrices.
Matrices (and, accordingly, the mathematical section - matrix algebra) are important in applied mathematics, since they allow one to write down a significant part of mathematical models of objects and processes in a fairly simple form. The term "matrix" appeared in 1850. Matrices were first mentioned in ancient China, and later by Arab mathematicians.
Matrix A=A mn order m*n is called rectangular table of numbers containing m - rows and n - columns.
Matrix elements aij, for which i=j are called diagonal and form main diagonal.
For a square matrix (m=n), the main diagonal is formed by the elements a 11, a 22,..., a nn.
Matrix equality.
A=B, if the matrix orders A And B are the same and a ij =b ij (i=1,2,...,m; j=1,2,...,n)
Actions on matrices.
1. Matrix addition - element-wise operation
Matrix subtraction - element-wise operation
3. The product of a matrix and a number is an element-wise operation
4. Multiplication A*B matrices according to the rule row to column(the number of columns of matrix A must be equal to the number of rows of matrix B)
A mk *B kn =C mn and each element with ij matrices Cmn is equal to the sum of the products of the elements of the i-th row of matrix A and the corresponding elements of the j-th column of matrix B.
Let's show the operation of matrix multiplication using an example:
6. Transpose of matrix A. The transposed matrix is denoted by A T or A"
Rows and columns swapped
Example
Properties of operations on matrices
(A+B)+C=A+(B+C)
λ(A+B)=λA+λB
A(B+C)=AB+AC
(A+B)C=AC+BC
λ(AB)=(λA)B=A(λB)
A(BC)=(AB)C
Types of matrices
1. Rectangular: m And n- arbitrary positive integers
2. Square: m=n
3. Matrix row: m=1. For example, (1 3 5 7) - in many practical problems such a matrix is called a vector
4. Matrix column: n=1. For example
5. Diagonal matrix: m=n And a ij =0, If i≠j. For example
6. Identity matrix: m=n And
7. Zero matrix: a ij =0, i=1,2,...,m
j=1,2,...,n
8. Triangular matrix: all elements below the main diagonal are 0.
9. Square Matrix: m=n And a ij =a ji(i.e., equal elements are located in places symmetrical relative to the main diagonal), and therefore A"=A
For example,
Inverse matrix- such a matrix A−1, when multiplied by which the original matrix A results in the identity matrix E:
A square matrix is invertible if and only if it is non-singular, that is, its determinant is not equal to zero. For non-square matrices and singular matrices, there are no inverse matrices. However, it is possible to generalize this concept and introduce pseudoinverse matrices, similar to inverses in many properties.
Examples of solving systems of linear algebraic equations using the matrix method.
Let's look at the matrix method using examples. In some examples we will not describe in detail the process of calculating determinants of matrices.
Example.
Using the inverse matrix, find the solution to the system of linear equations
.
Solution.
In matrix form, the original system will be written as, where 
. Let's calculate the determinant of the main matrix and make sure that it is different from zero. Otherwise, we will not be able to solve the system using the matrix method. We have 
, therefore, for the matrix A the inverse matrix can be found. Thus, if we find the inverse matrix, then we define the desired solution of the SLAE as . So, the task has been reduced to constructing the inverse matrix. Let's find her.
The inverse matrix can be found using the following formula:
, where is the determinant of matrix A, is the transposed matrix of algebraic complements of the corresponding elements of matrix .
The concept of an inverse matrix exists only for square matrices, matrices “two by two”, “three by three”, etc.
Polar coordinates. In the polar coordinate system, the position of point M
M
RECTANGULAR COORDINATES IN SPACE
STRAIGHT
1. General equation of a straight line. Any equation of the first degree with respect to x and y, i.e. an equation of the form:
(1) Ax+Bu+C=0 called. communities by the equation of the straight line ( + ≠0), A, B, C - CONSTANT COEFFICIENTS.
SECOND ORDER CURVES
1. Circle. A circle is a set of points in a plane, equidistant -
equidistant from a given point (center). If r is the radius of the circle, and point C (a; b) is its center, then the equation of the circle has the form:
Hyperbola. A hyperbola is a set of points on a plane, the absolute
the magnitude of the difference in distances to two given points, called fo-
pieces, there is a constant value (it is denoted by 2a), and this constant is less than the distance between the foci. If we place the foci of the hyperbola at the points F1 (c; 0) and F2(- c; 0), we obtain the canonical equation of the hyperbola 
ANALYTICAL GEOMETRY IN SPACE
FLAT AND STRAIGHT
plane, called the normal vector.
Second order surface
Second order surface- geometric locus of points in three-dimensional space whose rectangular coordinates satisfy an equation of the form
in which at least one of the coefficients , , , , , is different from zero.
Types of second order surfaces
Cylindrical surfaces
The surface is called cylindrical surface with generatrix, if for any point of this surface the straight line passing through this point parallel to the generatrix belongs entirely to the surface.
Theorem (about the equation of a cylindrical surface).
If in some Cartesian rectangular coordinate system the surface has the equation , then it is a cylindrical surface with a generatrix parallel to the axis.
A curve defined by an equation in the plane is called guide cylindrical surface.
If the guide of a cylindrical surface is given by a second-order curve, then such a surface is called cylindrical surface of the second order .
| Elliptical Cylinder: | Parabolic cylinder: | Hyperbolic cylinder: |
 |  |
|
 |  |  |
| A pair of matching lines: | Pair of coincident planes: | Pair of intersecting planes: |
 |  |
Conical surfaces
Conical surface.
Main article:Conical surface
The surface is called conical surface with apex at point, if for any point of this surface the straight line passing through and belongs entirely to this surface.
The function is called homogeneous order, if the following is true:
Theorem (on the equation of a conical surface).
If in some Cartesian rectangular coordinate system the surface is given by the equation 
, where is a homogeneous function, then is a conical surface with a vertex at the origin.
If a surface is defined by a function that is a homogeneous algebraic polynomial of the second order, then it is called conical surface of the second order .
· The canonical equation of a second order cone has the form:
Surfaces of revolution]
The surface is called surface of rotation around an axis, if for any point 
of this surface a circle passing through this point in a plane with center at and radius 
, belongs entirely to this surface.
Theorem (about the equation of a surface of revolution).
If in some Cartesian rectangular coordinate system the surface is given by the equation, then is a surface of rotation around the axis.
| Ellipsoid: | Single-sheet hyperboloid: | Two-sheet hyperboloid: | Elliptical paraboloid: |
 |  |  |  |
 |  |  |  |
In case , the surfaces listed above are surfaces of revolution.
Elliptical paraboloid
The equation of an elliptic paraboloid is
If , then an elliptic paraboloid is a surface of revolution formed by the rotation of a parabola whose parameter 
, around a vertical axis passing through the vertex and focus of a given parabola.
The intersection of an elliptic paraboloid with a plane is an ellipse.
The intersection of an elliptic paraboloid with a plane or is a parabola.
Hyperbolic paraboloid]
Hyperbolic paraboloid.
The equation of a hyperbolic paraboloid has the form
The intersection of a hyperbolic paraboloid with a plane is a hyperbola.
The intersection of a hyperbolic paraboloid with a plane or is a parabola.
Due to its geometric similarity, a hyperbolic paraboloid is often called a “saddle”.
Central surfaces
If the center of a second-order surface exists and is unique, then its coordinates can be found by solving the system of equations:
Thus, the sign that is assigned to the minor of the corresponding element of the determinant is determined by the following table:
In the above equality expressing the third order determinant,
on the right side there is the sum of the products of the elements of the 1st row of the determinant and their algebraic complements.
Theorem 1. The third-order determinant is equal to the sum of the products
elements of any of its rows or columns into their algebraic complements.
This theorem allows you to calculate the value of the determinant, revealing it according to
elements of any of its rows or columns.
Theorem 2. Sum of products of elements of any row (column)
the determinant of algebraic complements of elements of another row (column) is equal to zero.
Properties of determinants.
1°. The determinant will not change if the rows of the determinant are replaced by the column
tsami, and the columns are the corresponding rows.
2°. The common factor of the elements of any row (or column) can
be taken beyond the determinant sign.
3°. If the elements of one row (column) of the determinant, respectively
are equal to the elements of another row (column), then the determinant is equal to zero.
4°. When rearranging two rows (columns), the determinant changes sign to
opposite.
5°. The determinant will not change if the elements of the same row (column)
add the corresponding elements of another row (column), multiplied by the same number (the theorem on the linear combination of parallel series of the determinant).
Solving a system of three linear equations in three unknowns.
found using Cramer's formulas
It is assumed that D ≠0 (if D = 0, then the original system is either uncertain or inconsistent).
If the system is homogeneous, i.e., has the form 
and its determinant is nonzero, then it has a unique solution x = 0,
If the determinant of a homogeneous system is equal to zero, then the system is reduced
either to two independent equations (the third is their consequence), or to
one equation (the other two are its consequences). First case
occurs when among the minors of the determinant of a homogeneous system there is
at least one is different from zero, the second is when all the minors of this determinant are equal to zero. In both cases, a homogeneous system has an infinite number of solutions.
Calculate third order determinant
KOSTROMA BRANCH OF THE MILITARY UNIVERSITY OF RCB PROTECTION
Department of Automation of Troop Control
For teachers only
"I approve"
Head of Department No. 9
Colonel YAKOVLEV A.B.
"____"______________ 2004
Associate Professor A.I. SMIRNOVA
"QUALIFIERS.
SOLUTION OF SYSTEMS OF LINEAR EQUATIONS"
LECTURE No. 2 / 1
Discussed at department meeting No. 9
"____"___________ 2004
Protocol No.___________
Kostroma, 2004.
Introduction
1. Second and third order determinants.
2. Properties of determinants. Decomposition theorem.
3. Cramer's theorem.
Conclusion
Literature
1. V.E. Schneider et al., A Short Course in Higher Mathematics, Volume I, Ch. 2, paragraph 1.
2. V.S. Shchipachev, Higher Mathematics, chapter 10, paragraph 2.
INTRODUCTION
The lecture discusses determinants of the second and third orders and their properties. And also Cramer’s theorem, which allows you to solve systems of linear equations using determinants. Determinants are also used later in the topic “Vector Algebra” when calculating the vector product of vectors.
1st study question DETERMINANTS OF THE SECOND AND THIRD
ORDER
Consider a table of four numbers of the form
The numbers in the table are indicated by a letter with two indices. The first index indicates the row number, the second the column number.
DEFINITION 1.Second order determinant calledexpressionkind:
(1)Numbers A 11, …, A 22 are called elements of the determinant.
Diagonal formed by elements A 11 ; A 22 is called the main one, and the diagonal formed by the elements A 12 ; A 21 - side by side.
Thus, the second-order determinant is equal to the difference between the products of the elements of the main and secondary diagonals.
Note that the answer is a number.
EXAMPLES. Calculate:
Now consider a table of nine numbers, written in three rows and three columns:
DEFINITION 2. Third order determinant called an expression of the form:
Elements A 11; A 22 ; A 33 – form the main diagonal.
Numbers A 13; A 22 ; A 31 – form a side diagonal.
Let us depict schematically how the plus and minus terms are formed:
" + " " – "The plus includes: the product of the elements on the main diagonal, the remaining two terms are the product of the elements located at the vertices of triangles with bases parallel to the main diagonal.
The minus terms are formed according to the same scheme with respect to the secondary diagonal.
This rule for calculating the third-order determinant is called
Rule T reugolnikov.
EXAMPLES. Calculate using the triangle rule:
COMMENT. Determinants are also called determinants.
2nd study question PROPERTIES OF DETERMINANTS.
EXPANSION THEOREM
Property 1. The value of the determinant will not change if its rows are swapped with the corresponding columns.
.By revealing both determinants, we are convinced of the validity of the equality.
Property 1 establishes the equality of the rows and columns of the determinant. Therefore, we will formulate all further properties of the determinant for both rows and columns.
Property 2. When rearranging two rows (or columns), the determinant changes its sign to the opposite one, maintaining its absolute value.
.Property 3. Common factor of row elements(or column)can be taken out as a determinant sign.
.Property 4. If the determinant has two identical rows (or columns), then it is equal to zero.
This property can be proven by direct verification, or you can use property 2.
Let us denote the determinant by D. When two identical first and second rows are rearranged, it will not change, but according to the second property it must change sign, i.e.
D = - DÞ 2 D = 0 ÞD = 0.
Property 5. If all elements of a string(or column)are equal to zero, then the determinant is equal to zero.
This property can be considered as a special case of property 3 when
Property 6. If the elements of two lines(or columns)the determinants are proportional, then the determinant is equal to zero.
.Can be proven by direct verification or using properties 3 and 4.
Property 7. The value of the determinant will not change if the corresponding elements of another row (or column) are added to the elements of a row (or column), multiplied by the same number.
.Proved by direct verification.
The use of these properties can in some cases facilitate the process of calculating determinants, especially of the third order.
For what follows we will need the concepts of minor and algebraic complement. Let's consider these concepts to define the third order.
DEFINITION 3. Minor of a given element of a third-order determinant is called a second-order determinant obtained from a given element by crossing out the row and column at the intersection of which the given element stands.
Element minor Aij denoted by Mij. So for the element A 11 minor
It is obtained by crossing out the first row and first column in the third-order determinant.
DEFINITION 4. Algebraic complement of the element of the determinant they call it minor multiplied by(-1)k, Wherek- the sum of the row and column numbers at the intersection of which this element stands.
Algebraic complement of an element Aij denoted by Aij.
Thus, Aij =
.Let us write down the algebraic additions for the elements A 11 and A 12.
. .It is useful to remember the rule: the algebraic complement of an element of a determinant is equal to its signed minor plus, if the sum of the row and column numbers in which the element appears is even, and with a sign minus, if this amount odd.
EXAMPLE. Find minors and algebraic complements for the elements of the first row of the determinant:
It is clear that minors and algebraic complements can differ only in sign.
Let us consider without proof an important theorem - determinant expansion theorem.
EXPANSION THEOREM
The determinant is equal to the sum of the products of the elements of any row or column and their algebraic complements.
Using this theorem, we write the expansion of the third-order determinant along the first line.
.In expanded form:
.The last formula can be used as the main one when calculating the third-order determinant.
The expansion theorem allows us to reduce the calculation of the third-order determinant to the calculation of three second-order determinants.
The decomposition theorem provides a second way to calculate third-order determinants.
EXAMPLES. Calculate the determinant using the expansion theorem.
Practical work
“Solving systems of linear equations of the third order using the Cramer method”
Goals of work:
expand the understanding of methods for solving SLEs and work out the algorithm for solving SLEs using the Kramor method;
develop students’ logical thinking, the ability to find a rational solution to a problem;
to cultivate in students accuracy and culture of written mathematical speech when they formulate their solutions.
Basic theoretical material.
Cramer's method. Application for systems of linear equations.
A system of N linear algebraic equations (SLAEs) with unknowns is given, the coefficients of which are the elements of the matrix and the free terms are numbers
The first index next to the coefficients indicates in which equation the coefficient is located, and the second - in which of the unknowns it is found. 
If the matrix determinant is not zero
then the system of linear algebraic equations has a unique solution. The solution to a system of linear algebraic equations is such an ordered set of numbers that transforms each of the equations of the system into a correct equality. If the right-hand sides of all equations of the system are equal to zero, then the system of equations is called homogeneous. In the case when some of them are different from zero – heterogeneous 
If a system of linear algebraic equations has at least one solution, then it is called compatible, otherwise it is called incompatible. If the solution to the system is unique, then the system of linear equations is called definite. In the case where the solution to a joint system is not unique, the system of equations is called indeterminate. Two systems of linear equations are called equivalent (or equivalent) if all solutions of one system are solutions of the second, and vice versa. We obtain equivalent (or equivalent) systems using equivalent transformations. 
Equivalent transformations of SLAEs
1) rearrangement of equations;
2) multiplication (or division) of equations by a non-zero number;
3) adding another equation to some equation, multiplied by an arbitrary non-zero number.
The solution to the SLAE can be found in different ways, for example, using Cramer’s formulas (Cramer’s method)
Cramer's theorem. If the determinant of a system of linear algebraic equations with unknowns is nonzero, then this system has a unique solution, which is found using Cramer’s formulas: 
- determinants formed by replacing the th column with a column of free terms.
If , and at least one of them is different from zero, then the SLAE has no solutions. If 
, then the SLAE has many solutions.
A system of three linear equations with three unknowns is given. Solve the system using Cramer's method
Solution. 
Let's find the determinant of the coefficient matrix for unknowns
Since , then the given system of equations is consistent and has a unique solution. Let's calculate the determinants:
Using Cramer's formulas we find the unknowns
So 
the only solution to the system.
A system of four linear algebraic equations is given. Solve the system using Cramer's method.
Let's find the determinant of the coefficient matrix for the unknowns. To do this, let's expand it along the first line.
Let's find the components of the determinant:
Let's substitute the found values into the determinant
Determinant, therefore the system of equations is consistent and has a unique solution. Let's calculate the determinants using Cramer's formulas:
Evaluation criteria:
The work is graded “3” if: one of the systems is completely and correctly solved independently.
The work is graded “4” if: any two systems are completely and correctly solved independently.
The work is graded “5” if: three systems are completely and correctly solved independently.
Solving systems of linear algebraic equations (SLAEs) is undoubtedly the most important topic in a linear algebra course. A huge number of problems from all branches of mathematics come down to solving systems of linear equations. These factors explain the reason for this article. The material of the article is selected and structured so that with its help you can
- choose the optimal method for solving your system of linear algebraic equations,
- study the theory of the chosen method,
- solve your system of linear equations by considering detailed solutions to typical examples and problems.
Brief description of the article material.
First, we give all the necessary definitions, concepts and introduce notations.
Next, we will consider methods for solving systems of linear algebraic equations in which the number of equations is equal to the number of unknown variables and which have a unique solution. Firstly, we will focus on Cramer’s method, secondly, we will show the matrix method for solving such systems of equations, and thirdly, we will analyze the Gauss method (the method of sequential elimination of unknown variables). To consolidate the theory, we will definitely solve several SLAEs in different ways.
After this, we will move on to solving systems of linear algebraic equations of general form, in which the number of equations does not coincide with the number of unknown variables or the main matrix of the system is singular. Let us formulate the Kronecker-Capelli theorem, which allows us to establish the compatibility of SLAEs. Let us analyze the solution of systems (if they are compatible) using the concept of a basis minor of a matrix. We will also consider the Gauss method and describe in detail the solutions to the examples.
We will definitely dwell on the structure of the general solution of homogeneous and inhomogeneous systems of linear algebraic equations. Let us give the concept of a fundamental system of solutions and show how the general solution of a SLAE is written using the vectors of the fundamental system of solutions. For a better understanding, let's look at a few examples.
In conclusion, we will consider systems of equations that can be reduced to linear ones, as well as various problems in the solution of which SLAEs arise.
Page navigation.
Definitions, concepts, designations.
We will consider systems of p linear algebraic equations with n unknown variables (p can be equal to n) of the form
Unknown variables, - coefficients (some real or complex numbers), - free terms (also real or complex numbers).
This form of recording SLAE is called coordinate.
IN matrix form writing this system of equations has the form,
Where 
- the main matrix of the system, - a column matrix of unknown variables, - a column matrix of free terms.
If we add a matrix-column of free terms to matrix A as the (n+1)th column, we get the so-called extended matrix systems of linear equations. Typically, an extended matrix is denoted by the letter T, and the column of free terms is separated by a vertical line from the remaining columns, that is, 
Solving a system of linear algebraic equations called a set of values of unknown variables that turns all equations of the system into identities. The matrix equation for given values of the unknown variables also becomes an identity.
If a system of equations has at least one solution, then it is called joint.
If a system of equations has no solutions, then it is called non-joint.
If a SLAE has a unique solution, then it is called certain; if there is more than one solution, then – uncertain.
If the free terms of all equations of the system are equal to zero 
, then the system is called homogeneous, otherwise - heterogeneous.
Solving elementary systems of linear algebraic equations.
If the number of equations of a system is equal to the number of unknown variables and the determinant of its main matrix is not equal to zero, then such SLAEs will be called elementary. Such systems of equations have a unique solution, and in the case of a homogeneous system, all unknown variables are equal to zero.
We started studying such SLAEs in high school. When solving them, we took one equation, expressed one unknown variable in terms of others and substituted it into the remaining equations, then took the next equation, expressed the next unknown variable and substituted it into other equations, and so on. Or they used the addition method, that is, they added two or more equations to eliminate some unknown variables. We will not dwell on these methods in detail, since they are essentially modifications of the Gauss method.
The main methods for solving elementary systems of linear equations are the Cramer method, the matrix method and the Gauss method. Let's sort them out.
Solving systems of linear equations using Cramer's method.
Suppose we need to solve a system of linear algebraic equations 
in which the number of equations is equal to the number of unknown variables and the determinant of the main matrix of the system is different from zero, that is, .
Let be the determinant of the main matrix of the system, and 
- determinants of matrices that are obtained from A by replacement 1st, 2nd, …, nth column respectively to the column of free members: 
With this notation, unknown variables are calculated using the formulas of Cramer’s method as 
. This is how the solution to a system of linear algebraic equations is found using Cramer's method.
Example.
Cramer's method 
.
Solution.
The main matrix of the system has the form 
. Let's calculate its determinant (if necessary, see the article): 
Since the determinant of the main matrix of the system is nonzero, the system has a unique solution that can be found by Cramer’s method.
Let's compose and calculate the necessary determinants 
(we obtain the determinant by replacing the first column in matrix A with a column of free terms, the determinant by replacing the second column with a column of free terms, and by replacing the third column of matrix A with a column of free terms): 
Finding unknown variables using formulas 
:
Answer:
The main disadvantage of Cramer's method (if it can be called a disadvantage) is the complexity of calculating determinants when the number of equations in the system is more than three.
Solving systems of linear algebraic equations using the matrix method (using an inverse matrix).
Let a system of linear algebraic equations be given in matrix form, where the matrix A has dimension n by n and its determinant is nonzero.
Since , matrix A is invertible, that is, there is an inverse matrix. If we multiply both sides of the equality by the left, we get a formula for finding a matrix-column of unknown variables. This is how we obtained a solution to a system of linear algebraic equations using the matrix method.
Example.
Solve system of linear equations matrix method.
Solution.
Let's rewrite the system of equations in matrix form: 
Because
then the SLAE can be solved using the matrix method. Using the inverse matrix, the solution to this system can be found as 
.
Let's construct an inverse matrix using a matrix from algebraic additions of elements of matrix A (if necessary, see the article): 
It remains to calculate the matrix of unknown variables by multiplying the inverse matrix 
to a matrix-column of free members (if necessary, see the article): 
Answer:
or in another notation x 1 = 4, x 2 = 0, x 3 = -1.
The main problem when finding solutions to systems of linear algebraic equations using the matrix method is the complexity of finding the inverse matrix, especially for square matrices of order higher than third.
Solving systems of linear equations using the Gauss method.
Suppose we need to find a solution to a system of n linear equations with n unknown variables 
the determinant of the main matrix of which is different from zero.
The essence of the Gauss method consists of sequential exclusion of unknown variables: first, x 1 is excluded from all equations of the system, starting from the second, then x 2 is excluded from all equations, starting from the third, and so on, until only the unknown variable x n remains in the last equation. This process of transforming system equations to sequentially eliminate unknown variables is called direct Gaussian method. After completing the forward stroke of the Gaussian method, x n is found from the last equation, using this value from the penultimate equation, x n-1 is calculated, and so on, x 1 is found from the first equation. The process of calculating unknown variables when moving from the last equation of the system to the first is called inverse of the Gaussian method.
Let us briefly describe the algorithm for eliminating unknown variables.
We will assume that , since we can always achieve this by rearranging the equations of the system. Let's eliminate the unknown variable x 1 from all equations of the system, starting with the second. To do this, to the second equation of the system we add the first, multiplied by , to the third equation we add the first, multiplied by , and so on, to the nth equation we add the first, multiplied by . The system of equations after such transformations will take the form 
where and 
.
We would have arrived at the same result if we had expressed x 1 in terms of other unknown variables in the first equation of the system and substituted the resulting expression into all other equations. Thus, the variable x 1 is excluded from all equations, starting from the second.
Next, we proceed in a similar way, but only with part of the resulting system, which is marked in the figure 
To do this, to the third equation of the system we add the second, multiplied by , to the fourth equation we add the second, multiplied by , and so on, to the nth equation we add the second, multiplied by . The system of equations after such transformations will take the form 
where and 
. Thus, the variable x 2 is excluded from all equations, starting from the third.
Next, we proceed to eliminating the unknown x 3, while we act similarly with the part of the system marked in the figure 
So we continue the direct progression of the Gaussian method until the system takes the form 
From this moment we begin the reverse of the Gaussian method: we calculate x n from the last equation as , using the obtained value of x n we find x n-1 from the penultimate equation, and so on, we find x 1 from the first equation.
Example.
Solve system of linear equations Gauss method.
Solution.
Let us exclude the unknown variable x 1 from the second and third equations of the system. To do this, to both sides of the second and third equations we add the corresponding parts of the first equation, multiplied by and by, respectively: 
Now we eliminate x 2 from the third equation by adding to its left and right sides the left and right sides of the second equation, multiplied by: 
This completes the forward stroke of the Gauss method; we begin the reverse stroke.
From the last equation of the resulting system of equations we find x 3: 
From the second equation we get .
From the first equation we find the remaining unknown variable and thereby complete the reverse of the Gauss method.
Answer:
X 1 = 4, x 2 = 0, x 3 = -1.
Solving systems of linear algebraic equations of general form.
In general, the number of equations of the system p does not coincide with the number of unknown variables n:
Such SLAEs may have no solutions, have a single solution, or have infinitely many solutions. This statement also applies to systems of equations whose main matrix is square and singular.
Kronecker–Capelli theorem.
Before finding a solution to a system of linear equations, it is necessary to establish its compatibility. The answer to the question when SLAE is compatible and when it is inconsistent is given by Kronecker–Capelli theorem:
In order for a system of p equations with n unknowns (p can be equal to n) to be consistent, it is necessary and sufficient that the rank of the main matrix of the system be equal to the rank of the extended matrix, that is, Rank(A)=Rank(T).
Let us consider, as an example, the application of the Kronecker–Capelli theorem to determine the compatibility of a system of linear equations.
Example.
Find out whether the system of linear equations has 
solutions.
Solution.
. Let's use the method of bordering minors. Minor of the second order 
different from zero. Let's look at the third-order minors bordering it: 
Since all the bordering minors of the third order are equal to zero, the rank of the main matrix is equal to two.
In turn, the rank of the extended matrix 
is equal to three, since the minor is of third order 
different from zero.
Thus, Rang(A), therefore, using the Kronecker–Capelli theorem, we can conclude that the original system of linear equations is inconsistent.
Answer:
The system has no solutions.
So, we have learned to establish the inconsistency of a system using the Kronecker–Capelli theorem.
But how to find a solution to an SLAE if its compatibility is established?
To do this, we need the concept of a basis minor of a matrix and a theorem about the rank of a matrix.
The minor of the highest order of the matrix A, different from zero, is called basic.
From the definition of a basis minor it follows that its order is equal to the rank of the matrix. For a non-zero matrix A there can be several basis minors; there is always one basis minor.
For example, consider the matrix 
.
All third-order minors of this matrix are equal to zero, since the elements of the third row of this matrix are the sum of the corresponding elements of the first and second rows.
The following second-order minors are basic, since they are non-zero 
Minors 
are not basic, since they are equal to zero.
Matrix rank theorem.
If the rank of a matrix of order p by n is equal to r, then all row (and column) elements of the matrix that do not form the chosen basis minor are linearly expressed in terms of the corresponding row (and column) elements forming the basis minor.
What does the matrix rank theorem tell us?
If, according to the Kronecker–Capelli theorem, we have established the compatibility of the system, then we choose any basis minor of the main matrix of the system (its order is equal to r), and exclude from the system all equations that do not form the selected basis minor. The SLAE obtained in this way will be equivalent to the original one, since the discarded equations are still redundant (according to the matrix rank theorem, they are a linear combination of the remaining equations).
As a result, after discarding unnecessary equations of the system, two cases are possible.
If the number of equations r in the resulting system is equal to the number of unknown variables, then it will be definite and the only solution can be found by the Cramer method, the matrix method or the Gauss method.
Example.
.
Solution.
Rank of the main matrix of the system 
is equal to two, since the minor is of second order 
different from zero. Extended Matrix Rank 
is also equal to two, since the only third order minor is zero 
and the second-order minor considered above is different from zero. Based on the Kronecker–Capelli theorem, we can assert the compatibility of the original system of linear equations, since Rank(A)=Rank(T)=2.
As a basis minor we take . It is formed by the coefficients of the first and second equations: 
The third equation of the system does not participate in the formation of the basis minor, so we exclude it from the system based on the theorem on the rank of the matrix: 
This is how we obtained an elementary system of linear algebraic equations. Let's solve it using Cramer's method: 
Answer:
x 1 = 1, x 2 = 2.
If the number of equations r in the resulting SLAE is less than the number of unknown variables n, then on the left sides of the equations we leave the terms that form the basis minor, and we transfer the remaining terms to the right sides of the equations of the system with the opposite sign.
The unknown variables (r of them) remaining on the left sides of the equations are called main.
Unknown variables (there are n - r pieces) that are on the right sides are called free.
Now we believe that free unknown variables can take arbitrary values, while the r main unknown variables will be expressed through free unknown variables in a unique way. Their expression can be found by solving the resulting SLAE using the Cramer method, the matrix method, or the Gauss method.
Let's look at it with an example.
Example.
Solve a system of linear algebraic equations 
.
Solution.
Let's find the rank of the main matrix of the system 
by the method of bordering minors. Let's take a 1 1 = 1 as a non-zero minor of the first order. Let's start searching for a non-zero minor of the second order bordering this minor: 
This is how we found a non-zero minor of the second order. Let's start searching for a non-zero bordering minor of the third order: 
Thus, the rank of the main matrix is three. The rank of the extended matrix is also equal to three, that is, the system is consistent.
We take the found non-zero minor of the third order as the basis one.
For clarity, we show the elements that form the basis minor: 
We leave the terms involved in the basis minor on the left side of the system equations, and transfer the rest with opposite signs to the right sides: 
Let's give the free unknown variables x 2 and x 5 arbitrary values, that is, we accept 
, where are arbitrary numbers. In this case, the SLAE will take the form 
Let us solve the resulting elementary system of linear algebraic equations using Cramer’s method: 
Hence, .
In your answer, do not forget to indicate free unknown variables.
Answer:
Where are arbitrary numbers.
Summarize.
To solve a system of general linear algebraic equations, we first determine its compatibility using the Kronecker–Capelli theorem. If the rank of the main matrix is not equal to the rank of the extended matrix, then we conclude that the system is incompatible.
If the rank of the main matrix is equal to the rank of the extended matrix, then we select a basis minor and discard the equations of the system that do not participate in the formation of the selected basis minor.
If the order of the basis minor is equal to the number of unknown variables, then the SLAE has a unique solution, which can be found by any method known to us.
If the order of the basis minor is less than the number of unknown variables, then on the left side of the system equations we leave the terms with the main unknown variables, transfer the remaining terms to the right sides and give arbitrary values to the free unknown variables. From the resulting system of linear equations we find the main unknown variables using the Cramer method, the matrix method or the Gauss method.
Gauss method for solving systems of linear algebraic equations of general form.
The Gauss method can be used to solve systems of linear algebraic equations of any kind without first testing them for consistency. The process of sequential elimination of unknown variables makes it possible to draw a conclusion about both the compatibility and incompatibility of the SLAE, and if a solution exists, it makes it possible to find it.
From a computational point of view, the Gaussian method is preferable.
See its detailed description and analyzed examples in the article Gauss method for solving systems of general linear algebraic equations.
Writing a general solution to homogeneous and inhomogeneous linear algebraic systems using vectors of the fundamental system of solutions.
In this section we will talk about simultaneous homogeneous and inhomogeneous systems of linear algebraic equations that have an infinite number of solutions.
Let us first deal with homogeneous systems.
Fundamental system of solutions homogeneous system of p linear algebraic equations with n unknown variables is a collection of (n – r) linearly independent solutions of this system, where r is the order of the basis minor of the main matrix of the system.
If we denote linearly independent solutions of a homogeneous SLAE as X (1) , X (2) , …, X (n-r) (X (1) , X (2) , …, X (n-r) are columnar matrices of dimension n by 1) , then the general solution of this homogeneous system is represented as a linear combination of vectors of the fundamental system of solutions with arbitrary constant coefficients C 1, C 2, ..., C (n-r), that is, .
What does the term general solution of a homogeneous system of linear algebraic equations (oroslau) mean?
The meaning is simple: the formula specifies all possible solutions of the original SLAE, in other words, taking any set of values of arbitrary constants C 1, C 2, ..., C (n-r), using the formula we will obtain one of the solutions of the original homogeneous SLAE.
Thus, if we find a fundamental system of solutions, then we can define all solutions of this homogeneous SLAE as .
Let us show the process of constructing a fundamental system of solutions to a homogeneous SLAE.
We select the basis minor of the original system of linear equations, exclude all other equations from the system and transfer all terms containing free unknown variables to the right-hand sides of the system equations with opposite signs. Let's give the free unknown variables the values 1,0,0,...,0 and calculate the main unknowns by solving the resulting elementary system of linear equations in any way, for example, using the Cramer method. This will result in X (1) - the first solution of the fundamental system. If we give the free unknowns the values 0,1,0,0,…,0 and calculate the main unknowns, we get X (2) . And so on. If we assign the values 0.0,…,0.1 to the free unknown variables and calculate the main unknowns, we obtain X (n-r) . In this way, a fundamental system of solutions to a homogeneous SLAE will be constructed and its general solution can be written in the form .
For inhomogeneous systems of linear algebraic equations, the general solution is represented in the form , where is the general solution of the corresponding homogeneous system, and is the particular solution of the original inhomogeneous SLAE, which we obtain by giving the free unknowns the values 0,0,...,0 and calculating the values of the main unknowns.
Let's look at examples.
Example.
Find the fundamental system of solutions and the general solution of a homogeneous system of linear algebraic equations 
.
Solution.
The rank of the main matrix of homogeneous systems of linear equations is always equal to the rank of the extended matrix. Let's find the rank of the main matrix using the method of bordering minors. As a non-zero minor of the first order, we take element a 1 1 = 9 of the main matrix of the system. Let's find the bordering non-zero minor of the second order: 
A minor of the second order, different from zero, has been found. Let's go through the third-order minors bordering it in search of a non-zero one: 
All third-order bordering minors are equal to zero, therefore, the rank of the main and extended matrix is equal to two. Let's take . For clarity, let us note the elements of the system that form it: 
The third equation of the original SLAE does not participate in the formation of the basis minor, therefore, it can be excluded: 
We leave the terms containing the main unknowns on the right sides of the equations, and transfer the terms with free unknowns to the right sides:
Let us construct a fundamental system of solutions to the original homogeneous system of linear equations. The fundamental system of solutions of this SLAE consists of two solutions, since the original SLAE contains four unknown variables, and the order of its basis minor is equal to two. To find X (1), we give the free unknown variables the values x 2 = 1, x 4 = 0, then we find the main unknowns from the system of equations 
.
Let's solve it using Cramer's method: 
Thus, .
Now let's construct X (2) . To do this, we give the free unknown variables the values x 2 = 0, x 4 = 1, then we find the main unknowns from the system of linear equations 
.
Let's use Cramer's method again: 
We get.
So we got two vectors of the fundamental system of solutions and , now we can write down the general solution of a homogeneous system of linear algebraic equations: 
, where C 1 and C 2 are arbitrary numbers., are equal to zero. We will also take the minor as a basic one, eliminate the third equation from the system and move the terms with free unknowns to the right-hand sides of the system equations: 
To find, let’s give the free unknown variables the values x 2 = 0 and x 4 = 0, then the system of equations will take the form 
, from where we find the main unknown variables using Cramer’s method: 
We have 
, hence, 
where C 1 and C 2 are arbitrary numbers.
It should be noted that solutions to an indeterminate homogeneous system of linear algebraic equations generate linear space Solution.
The canonical equation of an ellipsoid in a rectangular Cartesian coordinate system has the form 
. Our task is to determine the parameters a, b and c. Since the ellipsoid passes through points A, B and C, then when substituting their coordinates into the canonical equation of the ellipsoid, it should turn into an identity. So we get a system of three equations: 
Let's denote 
, then the system will become a system of linear algebraic equations 
.
Let us calculate the determinant of the main matrix of the system: 
Since it is non-zero, we can find the solution using Cramer’s method:
). Obviously, x = 0 and x = 1 are the roots of this polynomial. Quotient from division 
on 
is . Thus, we have an expansion and the original expression takes the form 
.
Let's use the method of indefinite coefficients. 
Equating the corresponding coefficients of the numerators, we arrive at a system of linear algebraic equations 
. Its solution will give us the desired indefinite coefficients A, B, C and D.
Let's solve the system using the Gaussian method: 
Using the reverse of the Gaussian method, we find D = 0, C = -2, B = 1, A = 1.
We get, 
Answer:
.
Consider a homogeneous system of third order differential equations
Here x(t), y(t), z(t) are the required functions on the interval (a, b), and ij (i, j =1, 2, 3) are real numbers.
Let us write the original system in matrix form
,
Where 
We will look for a solution to the original system in the form
,
Where 
, C 1 , C 2 , C 3 are arbitrary constants.
To find the fundamental system of solutions, you need to solve the so-called characteristic equation 
This equation is a third order algebraic equation, therefore it has 3 roots. The following cases are possible:
1. The roots (eigenvalues) are real and distinct.
2. Among the roots (eigenvalues) there are complex conjugate ones, let
- real root
=
3. The roots (eigenvalues) are real. One of the roots is a multiple.
To figure out how to act in each of these cases, we will need:
Theorem 1.
Let be the pairwise distinct eigenvalues of matrix A, and let be their corresponding eigenvectors. Then
form a fundamental system of solutions to the original system.
Comment
.
Let be the real eigenvalue of matrix A (the real root of the characteristic equation), and let be the corresponding eigenvector.
= - complex eigenvalues of matrix A, - corresponding - eigenvector. Then
(Re - real part, Im - imaginary part)
form a fundamental system of solutions to the original system. (i.e. and = considered together)
Theorem 3.
Let be the root of the characteristic equation of multiplicity 2. Then the original system has 2 linearly independent solutions of the form 
,
where , are vector constants. If the multiplicity is 3, then there are 3 linearly independent solutions of the form
.
Vectors are found by substituting solutions (*) and (**) into the original system.
To better understand the method for finding solutions of the form (*) and (**), see the typical examples below.
Now let's look at each of the above cases in more detail.
1. Algorithm for solving homogeneous systems of third-order differential equations in the case of different real roots of the characteristic equation.
Given the system
1) We compose a characteristic equation
- real and distinct eigenvalues of the 9roots of this equation).
2) We build where 
3) We build where
- eigenvector of matrix A, corresponding to , i.e. - any system solution 
4) We build where
- eigenvector of matrix A, corresponding to , i.e. - any system solution 
5)
constitute a fundamental system of solutions. Next we write the general solution of the original system in the form
,
here C 1, C 2, C 3 are arbitrary constants,
,
or in coordinate form 
Let's look at a few examples:
Example 1.
2) Find 
3) We find 
4) Vector functions 
or in coordinate notation 
Example 2.
1) We compose and solve the characteristic equation: 
2) Find 
3) We find 
4) Find 
5) Vector functions 
form a fundamental system. The general solution has the form 
or in coordinate notation 
2. Algorithm for solving homogeneous systems of third-order differential equations in the case of complex conjugate roots of the characteristic equation.
- real root,
2) We build where
 |
3) We build
| - eigenvector of matrix A, corresponding to , i.e. satisfies the system |  |
Here Re is the real part
Im - imaginary part
4) constitute a fundamental system of solutions. Next we write down the general solution of the original system:
, Where
C 1, C 2, C 3 are arbitrary constants.
Example 1.
1) Compose and solve the characteristic equation 
2) We are building
3) We build
, Where
Let's reduce the first equation by 2. Then add the first equation multiplied by 2i to the second equation, and subtract the first one multiplied by 2 from the third equation. 
Further 
Hence, 
4) - fundamental system of solutions. Let us write down the general solution of the original system: 
Example 2.
1) We compose and solve the characteristic equation 
2) We are building
(i.e., and considered together), where
Multiply the second equation by (1-i) and reduce by 2. 
Hence, 
3)
General solution of the original system
or 
2. Algorithm for solving homogeneous systems of third-order differential equations in the case of multiple roots of the characteristic equation.
We compose and solve the characteristic equation
There are two possible cases: 
Consider case a) 1), where
| - eigenvector of matrix A, corresponding to , i.e. satisfies the system | |
2) Let us refer to Theorem 3, from which it follows that there are two linearly independent solutions of the form 
,
where , are constant vectors. Let's take them for .
3) - fundamental system of solutions. Next we write down the general solution of the original system:
Consider case b):
1) Let us refer to Theorem 3, from which it follows that there are three linearly independent solutions of the form
,
where , , are constant vectors. Let's take them for .
2) - fundamental system of solutions. Next we write down the general solution of the original system.
To better understand how to find solutions of the form (*), consider several typical examples.
Example 1.
We compose and solve the characteristic equation: 
We have case a)
1) We build 
, Where
From the second equation we subtract the first:
? The third line is similar to the second, we cross it out. Subtract the second from the first equation: 
2) = 1 (multiples of 2)
According to T.3, this root must correspond to two linearly independent solutions of the form .
Let's try to find all linearly independent solutions for which, i.e. solutions of the form
.
Such a vector will be a solution if and only if is the eigenvector corresponding to =1, i.e.
, or 
, the second and third lines are similar to the first, throw them out. 
The system has been reduced to one equation. Consequently, there are two free unknowns, for example, and . Let's first give them the values 1, 0; then the values 0, 1. We get the following solutions: 
.
Hence, 
.
3) - fundamental system of solutions. It remains to write down the general solution of the original system: 
. .. Thus, there is only one solution of the form Let's substitute X 3 into this system: Cross out the third line (it is similar to the second). The system is consistent (has a solution) for any c. Let c=1. 
or 
