The concept of the nth order determinant. Determinants of nth order; minors and algebraic complements. Properties and calculation of nth order determinants

DETERMINANTS. MATRICES

1. The concept of an nth order determinant.

2. Methods for calculating determinants of the 2nd and 3rd orders.

3. Laplace's theorem.

4. Matrices and their types. Actions on matrices.

5. Inverse matrix.

6. Matrix rank.

1. The concept of an nth order determinant.

The nth order determinant is written as a square table containing n rows and n columns:

The numbers a ij are the elements of the determinant, i is the row number, j is the column number, n is the order of the determinant.

The diagonal of the determinant, consisting of elements with the same indices, is called main , and the other is called side .

A determinant of nth order is a number that is the algebraic sum of n! terms, each of which is the product of n elements, taken one from each row and from each column, and the sign of each term is determined by the elements included in its composition.

Basic properties of determinantsn - th order.

1. When replacing rows with columns, the value of the determinant does not change.

2. When two rows (columns) are rearranged, the determinant changes sign.

3. If all elements of any row (column) of the determinant are equal to zero, then the determinant is equal to zero.

4. If a determinant has two identical or proportional rows (columns), then such a determinant is equal to zero.

5. The common factor of all elements of a row (column) can be taken beyond the sign of the determinant.

6. The value of the determinant will not change if elements of another row (column) are added to the elements of a row (column), multiplied by the same number.

7. If the elements of a row (column) are a linear combination of the corresponding elements of two (or several) other rows (columns), then such a determinant is equal to zero.

2. Methods for calculating determinants of the 2nd and 3rd orders.

Size called determinant second order and denote .

Thus,

Determinant third order name the quantity

This formula is called Sarrus rule(rule of “triangles”) for calculating 3rd order determinants. To better remember the formula, you can create a Sarrus table by adding the first and second columns to the determinant. Then all terms will be the product of elements along the diagonals.

Examples: Calculate determinants:

3. Laplace's theorem.

Calculating determinants of higher orders directly is very difficult, therefore, to calculate them, the properties of determinants are used, as well as Laplace’s theorem, which allows one to reduce the order of a given determinant.

Let the determinant be given:

Let us cross out the i-th row and the j-th column in this determinant, at the intersection of which the element a ij is located. Then we obtain the determinant M ij

(n-1) – th order, which is called minor element a ij .

Algebraic complement A ij element a ij is called the minor of this element, taken with a sign (+) if the sum of indices i+j is an even number, and with a sign (-) if this sum is an odd number, i.e.

A ij = (-1) i + j M ij

Example. A third-order determinant is given

Find the minor and algebraic complement of element a 32.

Solution. ,

Laplace's theorem: The sum of the products of the elements of a row (column) by their corresponding algebraic additions equal to the determinant, i.e.

This theorem makes it possible to decompose the determinant into elements of some row or column and reduce its calculation to the calculation of determinants of lower order. In this case, the calculation of the determinant is greatly simplified if there are zeros among the elements of a certain row (column).

4. Matrices and their types. Actions on matrices.

A matrix of dimension kxn is called rectangular table numbers:

The numbers a ij are called its elements. In a compact form, the matrix can be written:, i=1, …, k, j=1, …, n. Matrices are denoted by capital letters A, B, C, ..., matrix elements are denoted by lowercase letters with double indexing.

Types of matrices.

The matrix is called square n-th order, if the number of rows is equal to the number of columns and is equal to n.

A matrix consisting of one row is called matrix-row.

A matrix consisting of one column is called matrix-column.

If we rearrange rows and columns in matrix A, we get a new matrix A T transposed to matrix A:

A matrix in which all elements are equal to 0 is called null.

A square matrix whose elements along the main diagonal are 1 and the rest are zero is called single matrix. It is designated by the letter E.

A square matrix of nth order is called degenerate (special), if the nth order determinant composed of its elements is equal to zero. If this determinant is different from zero, then the matrix is called non-degenerate (non-singular).

The two matrices are called equal, if their corresponding elements are identically equal.

Actions on matrices.

1. Addition (subtraction) of matrices.

Two matrices of the same dimension, i.e. matrices having the same number of rows and the same number of columns can be added (subtracted). In this case, the sum (difference) of two matrices is understood as a new matrix, the elements of which are equal to the sum (difference) of the corresponding elements of these matrices.

2. Multiplying a matrix by a number.

To multiply a matrix by a number, you need to multiply each element of this matrix by that number.

3. Matrix multiplication.

Two matrices can be multiplied only when the number of columns of the first matrix coincides with the number of rows of the second matrix.

The product of matrix A and matrix B is a new matrix C in which the element with ijj located at the intersection of the i-th row and the j-th column is equal to the sum of the products of the elements of the i-th row of matrix A and the elements of the j-th column of matrix B. Matrix C has as many rows as matrix A and as many columns as matrix B. The rule for multiplying matrices is called “row by column”.

Comment : matrix multiplication operation in general case not commutable, i.e. AB ≠ BA.

Example. Find the product of matrices A and B: C=AB,

Where , .

nth order determinant

The determinant or determinant of the nth order is a number written in the form

And calculated from these numbers(real or complex) - elements of the determinant

Schemes for calculating determinants of the 2nd and 3rd orders

Cramer's theorem.

Let (delta) be the determinant of the matrix systems A,a The (delta)i-determinant of the matrix is obtained from matrix A by replacing the j-th column of the columns of free numbers. Then, if (delta) is not equal to 0, then the system has a unique solution, defined in the formula:

1. The 2nd order determinant is calculated using the formula

2. The third-order determinant is calculated using the formula

There is a convenient scheme for calculating the third-order determinant (see Fig. 1 and Fig. 2).

Property of determinants

1. If any row (column) of the matrix consists of only zeros, then its determinant is 0.

2. If all elements of any row (column) of a matrix are multiplied by number (lambda), then its determinant will be multiplied by this number (lambda).

3.When a matrix is transposed, its determinant does not change.

Transpose-in mathematics, this is a transformation square matrix-replacing columns with rows or vice versa.

4.When two rows (columns) of a matrix are rearranged, its determinant changes sign to the opposite one.

5. If a square matrix contains two identical rows (columns), then its determinant is 0

6. If the elements of two rows (columns) of a matrix are proportional, then its determinant is 0

7. The sum of the products of the elements of any row (column) of a matrix with the algebraic complements of the elements of another row (column) of this matrix is equal to 0

8. The determinant of a matrix does not change if elements of another row (column), previously multiplied by the same number, are added to the elements of any row (column) of the matrix.

9. The sum of the products of the numbers b1,b2,...,bn by the algebraic complement of the elements of any row (column) is equal to the determinant of the matrix obtained from this replacement of the elements of this row (column) b1,b2,...bn.

10. The determinant of the product of two square matrices is equal to the product of their determinants |C|=|A|*|B|, where C=A*B;A and B-matrices of the nth order.

Considering the expanded expression for determinants

we note that each term includes as factors one element from each row and one from each column of the determinant, and all possible products of this type are included in the determinant with a plus or minus sign. This property is used as the basis for generalizing the concept of a determinant to square matrices of any order. Namely: the determinant of a square matrix of order, or, in short, the determinant of order, is the algebraic sum of all possible products of matrix elements, taken one from each row and one from each column, and the resulting products are equipped with plus and minus signs according to some well-defined rule. This rule is introduced

in a rather complex way, and we will not dwell on its formulation. It is important to note that it is established in such a way that the following most important basic property of the determinant is ensured:

1. When two rows are rearranged, the determinant changes sign to the opposite one.

For a determinant of 2nd and 3rd orders, this property can be easily verified by direct calculation. In the general case, it is proved on the basis of the rule of signs that we have not formulated here.

Determinants have a number of other remarkable properties that make it possible to successfully use determinants in a variety of theoretical and numerical calculations, despite the extreme cumbersomeness of the determinant: after all, the nth-order determinant contains, as is easy to see, terms, each term consists of factors and the terms are equipped signs according to some complex rule.

We move on to listing the main properties of determinants, without dwelling on their detailed proofs.

The first of these properties has already been formulated above.

2. The determinant does not change when its matrix is transposed, i.e., when replacing rows with columns while maintaining order.

The proof is based on a detailed study of the rules for placing signs in the terms of the determinant. This property makes it possible to transfer any statement concerning the rows of the determinant to the columns.

3. There is a determinant linear function from the elements of any of its rows (or columns). More details

where represent expressions that do not depend on the elements of the string.

This property clearly follows from the fact that each term contains one and only one factor from each, in particular, row.

Equality (5) is called the expansion of the determinant into the elements of the string, and the coefficients are called the algebraic complements of the elements in the determinant.

4. The algebraic complement of an element is equal, up to sign, to the so-called minor of the determinant, i.e., the determinant

the proportion of order obtained from a given by deleting a row and a column. To obtain the algebraic complement, the minor must be taken with the sign. Properties 3 and 4 reduce the calculation of the order determinant to the calculation of the order determinants

A number of interesting properties of determinants follow from the listed basic properties. Let's list some of them.

5. A determinant with two identical lines is equal to a bullet.

Indeed, if the determinant has two identical rows, then when they are rearranged, the determinant does not change, because the rows are identical, but at the same time, due to the first property, it changes its sign to the opposite. Therefore it is equal to zero.

The sum of the products of the elements of any row and the algebraic complements of another row is zero.

Indeed, such a sum is the result of the expansion of a determinant with two identical rows in one of them.

The common factor of the elements of any row can be taken out of the determinant sign.

This follows from property 3.

8. A determinant with two proportional rows is equal to zero.

It is enough to remove the proportionality factor, and we will get a determinant with two equal lines.

9. The determinant does not change if numbers proportional to the elements of another row are added to the elements of a row.

Indeed, by virtue of property 3, the transformed determinant: is equal to the sum of the original determinant of the determinant with two proportional rows, which is equal to zero.

The last property gives good remedy to calculate determinants. Using this property, you can, without changing the value of the determinant, transform its matrix so that in any row (or column) all elements except one are equal to zero. Then, having expanded the determinant to the elements of this row (column), we reduce the calculation of the determinant of order to the calculation of one determinant of order, namely, the algebraic complement of the only non-zero element of the selected row.

Consider square table A.

Definition. The nth order determinant is a number obtained from the elements of a given table according to the following rule:

1 .The determinant of the nth order is equal to the algebraic sum n! members.

Each term is the product of n-elements taken one from each row and each column of the table.

2 .The term is taken with a plus sign if the permutations formed by the first and second indices of the elements included in products of the same parity (either both even or odd) and with a minus sign in the opposite case.

The determinant is indicated by the symbol:

or briefly det A=.(determinant A)

According to definition = -.

Rule for calculating the 3rd order determinant:

Minors and algebraic complements

Let a determinant of nth order (n>1) be given

Definition 1. The minor of an element of the determinant of the nth order is the determinant of the (n-1)th order obtained from A by crossing out the i-th row and j-th column at the intersection of which the given element stands.

For example:

Definition 2. The algebraic complement of an element is the number

Basic properties of nth order determinants

1.On the equivalence of rows and columns.

The value of the nth order determinant does not change if its rows are replaced with corresponding columns.

2. If two rows (columns) of determinants are swapped, then the determinant will change sign to the opposite.

= k

If all elements of any row (or column) of a determinant have a common factor, then this common multiplier can be taken out as a determinant sign.

4. The value of a determinant is zero if all elements of any of its rows (or columns) are zeros.

5. A determinant with two proportional rows is equal to 0.

For example:

6. The value of the determinant will not change if the corresponding elements of another row, multiplied by the same number, are added to its elements of any row.

7. If the elements of any row i of the determinant are presented as the sum of two terms, then the determinant is equal to the sum of two determinants in which all lines except the i-th are the same as in the given determinant, and the i-th line of one determinant consists of the first terms, and the second of the second.

8. The determinant is equal to the sum of the products of all elements of any of its rows and their algebraic complements.

9. The sum of the products of all elements of any row of the determinant by the algebraic complements of the corresponding elements of another row is equal to zero.

For example:

Laplace's theorem

Theorem. Let k rows (or k columns) be arbitrarily chosen in the determinant d of order n, 1. Then the sum of the products of all minors of order k contained in the selected rows and their algebraic complements is equal to the determinant d.

Consequence. A special case of Laplace's theorem is the expansion of the determinant in a row or column. It allows you to represent the determinant of a square matrix as the sum of the products of the elements of any of its rows or columns and their algebraic complements.

Let be a square matrix of size . Let also be given some row number i or column number j of the matrix A. Then the determinant of A can be calculated using the following formulas:

Decomposition in the i-th row:

Decomposition along the jth row:

where is the algebraic complement to the minor located in row number i and column number j.

The statement is a special case of Laplace's theorem. It is enough to put k equal to 1 and select the th row, then the minors located in this row will be the elements themselves.

Examples for self-solution.

1. Find x from the equations and check by substituting the root into the determinant.

A); b)

Methods for calculating nth order determinants.

Let an ordered set be given n elements. Any arrangement n elements in a certain order is called rearrangement from these elements.

Since each element is determined by its number, we will say that given n natural numbers.

Number of different permutations from n numbers are equal to n!

If in some permutation of n numbers number i costs earlier j, But i > j, i.e. larger number stands before the smaller one, then they say that the pair i, j amounts to inversion.

Example 1. Determine the number of inversions in the permutation (1, 5, 4, 3, 2)

Solution.

The numbers 5 and 4, 5 and 3, 5 and 2, 4 and 3, 4 and 2, 3 and 2 form inversions. The total number of inversions in this permutation is 6.

The permutation is called even, if the total number of inversions in it is even, otherwise it is called odd. In the example discussed above, an even permutation is given.

Let some permutation be given..., i, …, j, … (*) . Conversion in which numbers i And j change places, and the rest remain in their places, is called transposition. After number transposition i And j in permutation (*) there will be a rearrangement..., j, …, i, ..., where all elements except i And j, remained in their places.

From any permutation from n numbers, you can go to any other permutation of these numbers using several transpositions.

Every transposition changes the parity of the permutation.

At n ≥ 2 number of even and odd permutations from n numbers are the same and equal.

Let M– ordered set of n elements. Any bijective transformation of a set M called substitutionnth degree.

Substitutions are written like this: https://pandia.ru/text/78/456/images/image005_119.gif" width="27" height="19"> and that's all ik are different.

Substitution called even, if both of its rows (permutations) have the same parity, i.e., either both even or both odd. Otherwise substitution called odd.

At n ≥ 2 number of even and odd substitutions nth degrees the same and equal to .

The determinant of a square matrix A of second order A= is the number equal to = a11a22–a12a21.

The determinant of a matrix is also called determinant. For the determinant of matrix A, the following notation is used: det A, ΔA.

Determinant square matrices A= third order call the number equal to │A│= a11a22a33+a12a23a31+a21a13a32-a13a22a31-a21a12a33-a32a23a11

Each term of the algebraic sum on the right side of the last formula is a product of matrix elements taken one and only one from each column and each row. To determine the sign of the product, it is useful to know the rule (it is called the triangle rule), schematically depicted in Fig. 1:

«+» «-»

https://pandia.ru/text/78/456/images/image012_64.gif" width="73" height="75 src=">.

Solution.

Let A be an nth order matrix with complex elements:

A=https://pandia.ru/text/78/456/images/image015_54.gif" width="112" height="27 src="> (1) ..gif" width="111" height="51"> (2) .

The determinant of the nth order, or the determinant of the square matrix A=(aij) for n>1, is the algebraic sum of all possible products of the form (1) , and the work (1) is taken with a “+” sign if the corresponding substitution (2) even, and with a “‑” sign if the substitution is odd.

Minor Mij element aij determinant is a determinant obtained from the original by deleting i th line and j- th column.

Algebraic complement Aij element aij the determinant is called a number Aij=(–1) i+ jMij, Where Mij – element minor aij.

Properties of determinants

1. The determinant does not change when replacing all rows with the corresponding columns (the determinant does not change when transposing).

2. When two rows (columns) are rearranged, the determinant changes sign.

3. A determinant with two identical (proportional) rows (columns) is equal to zero.

4. The factor common to all elements of a row (column) can be taken beyond the sign of the determinant.

5. The determinant will not change if the corresponding elements of another row (column) are added to the elements of a certain row (column), multiplied by the same number other than zero.

6. If all elements of a certain row (column) of a determinant are equal to zero, then it is equal to zero.

7. The determinant is equal to the sum of the products of the elements of any row (column) by their algebraic complements (the property of decomposing the determinant in a row (column)).

Let's look at some methods for calculating order determinants n .

1. If in an nth-order determinant at least one row (or column) consists of zeros, then the determinant is equal to zero.

2. Let some row in the nth order determinant contain non-zero elements. The calculation of the nth order determinant can be reduced in this case to the calculation of the n-1 order determinant. Indeed, using the properties of the determinant, you can make all elements of a row, except one, zero, and then expand the determinant along the specified row. For example, let us rearrange the rows and columns of the determinant so that in place a11 there was an element different from zero.

https://pandia.ru/text/78/456/images/image018_51.gif" width="32 height=37" height="37">.gif" width="307" height="101 src=">

Note that it is not necessary to rearrange rows (or columns). You can get zeros in any row (or column) of the determinant.

There is no general method for calculating determinants of order n, except for calculating the determinant of a given order directly by definition. To the determinant of one or another special type apply various methods calculations leading to simpler determinants.

3. Let's take it to triangular form. Using the properties of the determinant, we reduce it to the so-called triangular form, when all elements standing on one side of the main diagonal are equal to zero. The resulting triangular determinant is equal to the product of the elements on the main diagonal. If it is more convenient to get zeros on one side of the secondary diagonal, then it will be equal to the product of the elements of the secondary diagonal, taken with the sign https://pandia.ru/text/78/456/images/image022_48.gif" width="49" height= "37">.

Example 3. Calculate determinant by row expansion

https://pandia.ru/text/78/456/images/image024_44.gif" width="612" height="72">

Example 4. Calculate the fourth order determinant

https://pandia.ru/text/78/456/images/image026_45.gif" width="373" height="96 src=">.

2nd method(calculating the determinant by expanding it along the line):

Let us calculate this determinant by row expansion, having previously transformed it so that in some of its rows all elements except one become zero. To do this, add the first line of the determinant to the third. Then multiply the third column by (-5) and add it to the fourth column. We expand the transformed determinant along the third line. We reduce the third-order minor to triangular form relative to the main diagonal.

https://pandia.ru/text/78/456/images/image028_44.gif" width="202" height="121 src=">

Solution.

Let's subtract the second from the first line, the third from the second, etc., and finally, the last from the penultimate (the last line remains unchanged).

https://pandia.ru/text/78/456/images/image030_39.gif" width="445" height="126 src=">

The first determinant in the sum is triangular with respect to the main diagonal, so it is equal to the product of the diagonal elements, i.e. (n–1)n. We transform the second determinant in the sum by adding last line to all previous lines of the determinant. The determinant obtained from this transformation will be triangular with respect to the main diagonal, so it will be equal to the product of the diagonal elements, i.e. nn-1:

=(n–1)n+ (n–1)n + nn-1.

4. Calculation of the determinant using Laplace's theorem. If k rows (or columns) are selected in the determinant (1 £ k £ n-1), then the determinant is equal to the sum of the products of all k-th order minors located in the selected k rows (or columns) and their algebraic complements.

Example 6. Compute determinant

https://pandia.ru/text/78/456/images/image033_36.gif" width="538" height="209 src=">

INDIVIDUAL TASK No. 2

“CALCULATION OF NTH ORDER DETERMINANTS”

Option 1

Compute determinants

https://pandia.ru/text/78/456/images/image035_39.gif" width="114" height="94 src=">

Option 2

Compute determinants