What is a 2nd order determinant? Determinants and their properties

Determinant a square matrix is ​​a number that is calculated as follows:

a) If the order of a square matrix is ​​1, i.e. it consists of 1 number, then the determinant is equal to this number;

b) If the order of a square matrix is ​​2, i.e. it consists of 4 numbers, then the determinant is equal to the difference between the product of the elements of the main diagonal and the product of the elements of the secondary diagonal;

c) If the order of a square matrix is ​​3, i.e. it consists of 9 numbers, then the determinant is equal to the sum of the products of the elements of the main diagonal and two triangles parallel to this diagonal, from which the sum of the products of the elements of the secondary diagonal and two triangles parallel to this diagonal is subtracted.

Examples

Properties of determinants

1. The determinant will not change if the rows are replaced by columns and the columns by rows

1. A determinant having 2 identical series is equal to zero
2. The common factor of any row (row or column) of the determinant can be taken out of the sign of the determinant

4. When rearranging two parallel series, the determinant changes sign to the opposite one

5. If the elements of any series of a determinant are sums of two terms, then the determinant can be expanded into the sum of two corresponding determinants

6. The determinant will not change if the corresponding elements of a parallel series are added to the elements of one series, multiplied by any number

Minor element of the determinant and its algebraic complement

Minor element a IJ n-th order determinant is a n-1 order determinant obtained from the original one by crossing out the i-th row and j-th column

Algebraic complement of element a IJ determinant is its minor multiplied by (-1) i+ j

Example

Inverse matrix

The matrix is ​​called non-degenerate, if its determinant is not equal to zero, otherwise, the matrix is ​​called singular

The matrix is ​​called union, if it consists of the corresponding algebraic complements and is transposed

The matrix is ​​called reverse to a given matrix if their product is equal to the identity matrix of the same order as the given matrix

Theorem on the existence of an inverse matrix

Any non-singular matrix has an inverse equal to the union matrix divided by the determinant of this matrix

Algorithm for finding the inverse matrix A

1. Compute determinant

1. Transpose matrix

1. Construct a union matrix, calculate all algebraic complements of the transposed matrix

1. Use the formula:

Matrix minor is a determinant consisting of elements located at the intersection of selected k rows and k columns of a given matrix of size mxn

Matrix rank is the highest order of the matrix minor that is non-zero

Notation r(A), rangA

Rank is equal to the number of non-zero rows of the step matrix.

Example

Systems of linear equations.

A system of linear equations containing m equations and n unknowns is called a system of the form

where are the numbers a IJ - system coefficients, numbers b i - free terms

Matrix recording form systems of linear equations

System solution n values ​​of unknowns c 1, c 2,…, c n are called, when substituting them into the system, all equations of the system turn into true equalities. The solution to the system can be written as a column vector.

The system of equations is called joint, if it has at least one solution, and non-joint, if there are no solutions.

Kronecker–Capelli theorem

A LU system is consistent if and only if the rank of the main matrix is ​​equal to the rank of the extended matrix

Methods for solving a LU system

1. Gauss method(using elementary transformations, reduce the extended matrix to a step matrix and then to a canonical one)

Elementary transformations include:

Rearranging rows (columns)

Adding to one row (column) another, multiplied by a number other than 0.

Let's create an extended matrix:

Let's select the leading element in the first column and first row, element 1., and call it leading. The line containing the leading element will not change. Let's reset the elements under the main diagonal. To do this, add the first line to the second line, multiplied by (-2). Add the first line to the third line, multiplied by (-1), we get:

Let's swap the second and third lines. Mentally cross out the first column and the first row and continue the algorithm for the remaining matrix. To the third line we add the 2nd, multiplied by 5.

We brought the extended matrix to a stepped form. Returning to the equations of the system, starting from the last line and moving up, we determine the unknowns one by one.

2. Matrix method(AX=B, A -1 AX=A -1 B, X=A -1 B; the matrix inverse to the main matrix is ​​multiplied by the column of free terms)

3. Cramer's method.

The solution of the system is found by the formula:

Where is the determinant of the modified main matrix, in which the i-th column is changed to a column of free terms, and is the main determinant, consisting of the coefficients of the unknowns.

Vectors.

Vector is a directed segment

Any vector is given by length (modulus) and direction.

Designation: or

where A is the beginning of the vector, B is the end of the vector, and is the length of the vector.

Vector classification

Zero vector is a vector whose length is zero

Unit vector is a vector whose length is equal to one

Equal vectors– these are two vectors that have the same length and direction

Opposite vectors– these are two vectors whose lengths are equal and directions are opposite

Collinear vectors– these are two vectors that lie on the same line or on parallel lines

Codirectional vectors are two collinear vectors with the same direction

Oppositely directed vectors are two collinear vectors with opposite directions

Coplanar vectors are three vectors that lie in the same plane or on parallel planes

Rectangular system coordinates on a plane are two mutually perpendicular lines with a selected direction and origin, with the horizontal line called the abscissa axis, and the vertical line called the ordinate axis

For each point in a rectangular coordinate system we assign two numbers: abscissa and ordinate

Rectangular system coordinates in space are three mutually perpendicular straight lines with a chosen direction and origin, while the horizontal straight line directed towards us is called the abscissa axis, the horizontal straight line directed to the right of us is the ordinate axis, and the vertical straight line directed upward is called the applicate axis

For each point in a rectangular coordinate system we assign three numbers: abscissa, ordinate and applicate

2. DETERMINANTS AND CRAMER’S RULE

2.1. Second order determinants

Determinant second order matrix is ​​found as follows:

(2.1)
It is equal to the product of the elements of the main diagonal of the matrix minus the product of the elements of the second diagonal.

For example,


It should be emphasized once again that a matrix is ​​a table of numbers, while a determinant is a number determined through the elements of a square matrix.

Let us now consider a system of two linear equations with two unknowns:

Using the concept of a 2nd order determinant, the solution to this system can be written as:

(2.2)

It's there Cramer's rule solving a system of two linear equations with two unknowns, provided that 0.

Example 2.1. Solve a system of linear equations using Cramer's rule:

Solution . Let's find the determinants:



Historical information. Concept idea "determinant" could belong G. Leibniz(1646-1716), if he had developed and published his ideas regarding determinants, which he arrived at in 1693. Therefore, priority in developing a method of determinants for solving systems of linear equations belongs to G. Kramer(1704-1752), who published his research on this topic in 1750. However, Cramer did not build a full-fledged theory of determinants, and he also lacked a convenient notation. The first extensive study devoted to determinants was A. Vandermonde(1735-1796) in 1772. He gave a logical presentation of the theory of determinants and introduced the rule for decomposing a determinant using minors. A complete exposition of the theory of determinants was given only in 1812.
J. Binet(1786-1856) and O. Cauchy(1789-1858). Term "determinant" ("determinant") in its modern meaning was introduced by Cauchy (previously this term was used by K. Gauss to denote the discriminant of a quadratic form).

2.2. Third order determinants

(2.3)

Naturally, it is quite difficult to remember this formula. However, there are rules that make it easier to write out an expression for a 3rd order determinant.

Triangle rule : the three terms included in the original expression with a plus sign are products of elements of the main diagonal or triangles whose bases are parallel to this diagonal. The remaining three terms included with a minus sign are found in the same way, but relative to the second diagonal.

Sarrus rule : add the first and then the second column to the matrix on the right. Then the “positive” terms will be on lines parallel to the main diagonal, and the “negative” ones on lines parallel to the second diagonal.

2.3. Cramer's Rule

Using 3rd order determinants, the solution to 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's there 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:

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

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

Examination:

Therefore, the solution was found correctly. 

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

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

(2.5)

Where  – determinant of the main matrix,  i – matrix determinant, obtained from the main one, replacingith column column of free members.

Note that if =0, then Cramer’s rule does not apply. 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 about calculating determinants of higher orders.

2.4. Determinants of nth order

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

We get



Using the concept of algebraic complement we can formulate 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 a certain row (or column) by 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 over any row or column, we get n determinants ( n–1)th order. To have fewer such determinants, it is advisable to select 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 sorting them into some row or column. Typically, in such cases, select the column or row that has the most zeros. The selected row or column will be indicated by an arrow.



2.5. Basic properties of determinants

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

.

This property indicates the equality of the rows and columns of the determinant. In other words, any statement about the columns of a determinant is also 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 determinant sign.

For example,

Consequence. If all elements of a certain row (column) of a 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 product of matrices is equal to the product of the determinants of matrices:

2.6.

Elementary transformations The following transformations are called matrices: 1) multiplication of a row (column) by a number not equal to zero; 2) adding one row (column) to another; 3) rearrangement of two rows (columns).

Elementary transformation method is to use elementary transformations, taking into account the properties of the determinants, to reduce the matrix to a triangular form.

Example 2.5. Calculate the determinant using elementary transformations, bringing them to triangular form:



Example 2.6. Calculate the determinant:

.

Solution . Let's simplify this determinant and then calculate it:

. 
Example 2.7. Compute determinant
.

Solution . Method 1 Using elementary transformations of the matrix, taking into account the properties of the determinants, we will obtain zeros in any row or column, and then we will expand the resulting determinant along this row or column:

–6

2

-2


.
Method 2 .Using elementary transformations of the matrix, taking into account the properties of the determinants, we reduce the matrix to triangular form:

. 

Calculating determinants using elementary transformations, by reducing it to triangular form, is one of the most common methods. This is due to the fact that it is the main method for calculating determinants on a computer. More precisely, it is one of the modifications Gauss method , which is usually used when solving systems of linear equations.

Example 2.8. Calculate the determinant using the Gaussian method:

Solution. Consider the first column and select the row in it that contains 1. If there are no units, then you need to create this unit using elementary transformations: rearranging rows or columns, adding or subtracting them with each other, multiplying or dividing them by some number (taking into account, of course, the properties of determinants). Let's take the second row as a basis and use it to obtain zeros in the first column:

After this, we don’t pay any more attention to the first line. Let's look at the 2nd column.

The result is a triangular matrix. In order to calculate the determinant, all that remains is to multiply the matrix elements located on the main diagonal. Thus, we get the answer: –2(–1)(–1)1334 = –264. 

Practical lesson

Subject: Calculation of determinants.

Goals: h strengthen the concepts of determinants and their properties, to form and consolidate skills and abilities calculate determinants of the 2nd and 3rd orders; develop the ability to generalize acquired knowledge, conduct analysis and comparisons, promote the development of logical thinking; to cultivate in students a conscious attitude towards the learning process.

I. General theoretical principles

A second-order determinant is a number

A third-order determinant is a number

Properties of determinants

Property 1.
The determinant will not change if all rows are replaced by the corresponding columns and vice versa.

Property 2.
When any two rows or columns are swapped, the determinant changes sign.

Property 3.
A determinant is equal to zero if it has two equal rows (columns).

Property 4.
A factor common to all elements of a row or column can be taken out of the determinant sign.

Property 5.
If the corresponding elements of another row or column are added to the elements of a row or column, the determinant will not change.

Corollary from properties 4 and 5: If to the elements of a row or column we add the corresponding elements of another row or column, multiplied by a certain number, then the determinant will not change.

Security questions:

1.Give the definition of a matrix.
2. What does the symbol mean? ?
3. Which matrix is ​​called transposed with respect to matrix A?
4. What matrix is ​​called square of order n?
5. Define a 2nd order determinant.

6. Define a 3rd order determinant.

7. What is the determinant of a transposed matrix?

8. How will the value of the determinant change if 2 rows (columns) are swapped in the matrix?

9. Is it possible to take the common factor of a row or column out of the determinant sign?

10.What is the determinant if all elements of a certain row (column) are equal to 0?

11.What is the determinant equal to if it has two identical rows (columns)?

12. Formulate a rule for calculating the 2nd order determinant.

13. Formulate a rule for calculating the 3rd order determinant.

II . Formation of skills and abilities.

Example 1. You number the determinant : a) according to the triangle rule b) according to Sarrus’ rule;

c) by the method of expansion by elements of the first row

Solution:

b) add the first two columns and calculate the product of three elements along the main diagonal and parallel to it with a sign (+), and then along the secondary diagonal and parallel to it with a sign (-):

we get:

Example 2. Compute determinant in two ways: using the first row expansion and the triangle rule.

Solution:

Example 3. Calculate the determinant using the properties:

III .Reinforcement of the studied material.

No. 1. Calculate determinants:

№ 2. Solve the equations:

No. 4. Calculate determinants using the properties:

1 .
. 2.
. 3.
. 4 .
.

Literature

1. Pismenny, D. T. Lecture notes on higher mathematics: a complete course by D. T. Pismenny. – 9th ed. – M.: Iris-press, 2009. 608 p.: ill. – (Higher education).

2. Lungu, K. N. Collection of problems in higher mathematics. 1st year / K. N. Lungu, D. T. Pismenny, S. N. Fedin, Yu. A. Shevchenko. – 7th ed. – M.: Iris-press, 2008. 576 pp.: – (Higher education).

Topic 1. Matrices and systems

Matrix concept

Definition 1.Matrix

.

Here, a i j (i=1,2,...,m; j=1,2,...n) - matrix elements, i- line number, j m=n the matrix is ​​called square order matrix n.

i¹j are equal to zero, is called diagonal:

single

null and is denoted by θ.

- matrix row; - matrix column.

determinant(or determinant).

2nd order determinants

Definition 2. ABOUT second order limiter matrices , that is

. (3)

Other designations: , .

Thus, the concept of a determinant simultaneously presupposes a method for its calculation. The numbers are called elements of the determinant. The diagonal formed by the elements is called main and the elements - side

Example 1. The determinant of the matrix is ​​equal to

.

3rd order determinants

Definition 2. ABOUT third order limiter is the number denoted by the symbol

,

and defined by equality

Numbers - elements determinant. Elements form home diagonal, elements - side.

When calculating the determinant, in order to remember which terms on the right side of equality (4) are taken with the “+” sign and which with the “-” sign, use the symbolic rule of triangles (Sarrus’ rule):

With the “+” sign, the products of the elements of the main diagonal and the elements located at the vertices of triangles with bases parallel to the main diagonal are taken; followed by the “-” sign – the product of the elements of the secondary diagonal and the elements located at the vertices of triangles with bases parallel to the secondary diagonal.

Calculation of the determinant using the column assignment rule.

1. We assign the first and second columns sequentially to the right of the determinant.

2. We calculate the products of three elements diagonally from left to right, from top to bottom from A 11 to A 13 and take them with the “+” sign. Then we calculate the products of three elements diagonally from left to right, from bottom to top from A 31 to A 13 and take them with the “-” sign.

(-) (-) (-) (+) (+) (+)

Example 2. Calculate the determinant using the column assignment rule.

3. Determinants n-th order. Minors and algebraic complements. Calculation of determinants by row (column) expansion.

Let's consider the concept of a determinant n- no order. Determinant n- high order is the number associated with the matrix n- of a certain order and calculated according to a certain law.

,

here are the elements of the determinant. To show the rule by which the determinant is revealed n- order, let's look at some concepts.

Definition 4. Minor determinant element n-th order is called the determinant ( n- 1) order obtained by crossing out the row and column of the determinant at the intersection of which this element is located.

Definition 5. Algebraic complement some element of the determinant n The th order is called the minor of this element multiplied by , that is .

In a third-order determinant one can consider, for example,

, .

, .

Definition 6. Determinant n- of higher order is a number equal to the sum of the products of the elements of the first row of the determinant multiplied by their algebraic complements.

This rule for calculating the determinant is called expansion along the first row.

Theorem (about the expansion of the determinant). The determinant can be calculated by expanding over any row or column.

– the sum of the products of the elements of the 1st column by the algebraic complements of the 2nd column.

Example 3. Calculate fourth order determinant .

Solution. We multiply the third line by (-1) and add it to the fourth, then expand the determinant along the fourth line:

The third-order determinant was expanded along the first row.

Gauss method.

Gauss method is that the original system, by eliminating the unknown, is transformed to stepwise mind. In this case, transformations are performed on rows in the extended matrix, since transformations that exclude unknowns are equivalent to elementary transformations of matrix rows.

The Gaussian method consists of forward stroke And reverse motion. The direct approach of the Gauss method is to reduce the extended matrix of system (1) to a stepwise form by means of elementary transformations over the rows. After which the system is examined for consistency and certainty. Then the system of equations is reconstructed using the step matrix. The solution to this stepwise system of equations is the reverse of the Gaussian method, in which, starting from the last equation, unknowns with a large serial number are sequentially calculated and their values ​​are substituted into the previous equation of the system.

The study of the system at the end of the forward move is carried out according to the Kronecker-Capelli theorem by comparing the ranks of the system matrix A and the extended matrix A´. The following cases are possible.

1) If , then the system is inconsistent (according to the Kronecker-Capelli theorem).

2) If , then system (1) is definite, and vice versa (without proof).

3) If , then system (1) is uncertain, and vice versa (without proof).

Inequality does not hold, since matrix A is part of matrix A´, the inequality does not hold, since the number of columns of matrix A is equal n. Moreover, for a system with a square matrix, that is, if n = T, the equalities are equivalent to the fact that .

If the system is uncertain, that is, it is executed, then some of its unknowns are declared free, and the rest are expressed through them. The number of free unknowns is . When performing the reverse of the Gaussian method, if in the next equation, after substituting the previously found variables, more than one unknown remains, then any unknowns except one are declared free unknowns.

Let's look at the implementation of the Gauss method using examples.

Example 4. Solve the system of equations

Solution. Let's solve the system using the Gaussian method. Let's write out the extended matrix of the system and bring it to a stepwise form using elementary row transformations (direct motion).

~ ~ ~

~ ~ .

Therefore, the system is consistent and has a unique solution, i.e. is certain.

Let's create a stepwise system and solve it (reverse).

The check can be easily done by substitution.

Answer: .

Topic 2. Vector algebra.

Projection of a vector onto an axis.

Definition 2. Vector projection per axis l is a number equal to the length of the segment AB this axis, enclosed between the projections of the beginning and end of the vector, taken with a “+” sign, if the segment AB oriented (counting from A To IN) to the positive side of the axis l and the sign “-” - otherwise (see Fig. 2).

Designation: .

Theorem 1. The projection of a vector onto the axis is equal to the product of its modulus and the cosine of the angle between the vector and the positive direction of the axis (Fig. 3):

. (1)

Fig.3. Fig.4.

Proof. From (Fig. 3) we obtain . The direction of the segment coincides with the positive direction of the axis, therefore the equality is true. In the case of the opposite orientation (Fig. 4) we have . The theorem is proven.

Let's consider the properties of projections.

Property 1. The projection of the sum of two vectors and onto the axis is equal to the sum of their projections onto the same axis, that is.

Fig.5.

The proof in the case of one of the possible arrangements of vectors follows from Figure 5. Indeed, by definition 2

Property 1 is true for any finite number of terms of vectors.

Property 2. When a vector is multiplied by a number l, its projection is multiplied by this number

. (2)

Let us prove equality (2). When the vectors and form the same angle with the axis. By Theorem 1

When the vectors and form angles and with the axis, respectively. Theorem 1

For , we obtain the obvious equality

Corollary from properties 1 and 2. The projection of a linear combination of vectors is equal to the same linear combination of the projections of these vectors, i.e.

Topic 1. Matrices and systems

Matrix concept

Definition 1.Matrix size is a rectangular table of numbers or alphabetic expressions written in the form

.

Here, a i j (i=1,2,...,m; j=1,2,...n) - matrix elements, i- line number, j- column number. Matrices are usually denoted by capital letters of the Latin alphabet A, B, C, etc., as well as or . At m=n the matrix is ​​called square order matrix n.

A square matrix in which all elements have unequal indices i¹j are equal to zero, is called diagonal:

If all non-zero elements of a diagonal matrix are equal to one, then the matrix is ​​called single. The identity matrix is ​​usually denoted by the letter E.

A matrix whose elements are all zero is called null and is denoted by θ.

There are also matrices consisting of one row or one column.

- matrix row; - matrix column.

The numerical characteristic of a square matrix is determinant(or determinant).

Determinants of 2nd order and 3rd order, their properties.

2nd order determinants

Definition 2. ABOUT second order limiter matrices (or simply a second-order determinant) is a number denoted by a symbol and defined by the equality , that is

. (3)

Other designations: , .

To find the determinant of a matrix, you need to use formulas that are valid for determinants of 2nd and 3rd order.

Formula

Let a second-order matrix $ A = \begin(pmatrix) a_(11)&a_(12)\\a_(21)&a_(22) \end(pmatrix) $ be given. Then its determinant is calculated using the formula:

$$ \Delta = \begin(vmatrix) a_(11)&a_(12)\\a_(21)&a_(22) \end(vmatrix) = a_(11)\cdot a_(22) - a_(12)\ cdot a_(21) $$

From the product of the elements located on the main diagonal $ a_(11)\cdot a_(22) $, the product of the elements located on the secondary diagonal $ a_(12)\cdot a_(21) $ is subtracted. This rule is true only (!) for a 2nd order determinant.

If given a third order matrix $ A = \begin(pmatrix) a_(11)&a_(12)&a_(13)\\a_(21)&a_(22)&a_(23)\\a_(31)&a_(32)&a_ (33) \end(pmatrix) $, then its determinant should be calculated using the formula:

$$ \Delta = \begin(vmatrix) a_(11)&a_(12)&a_(13)\\a_(21)&a_(22)&a_(23)\\a_(31)&a_(32)&a_(33) \end(vmatrix) = $$

$$ = a_(11)a_(22)a_(33) + a_(12)a_(23)a_(31)+a_(21)a_(32)a_(13) - a_(13)a_(22) a_(31)-a_(23)a_(32)a_(11)-a_(12)a_(21)a_(33) $$

Examples of solutions