Determinant of the original matrix. Determinants of square matrices

Determinants and their properties. Rearrangement numbers 1, 2,..., n is any arrangement of these numbers in a certain order. In elementary algebra it is proven that the number of all permutations that can be formed from n numbers is 12...n = n!. For example, from three numbers 1, 2, 3 you can form 3!=6 permutations: 123, 132, 312, 321, 231, 213. They say that in this permutation the numbers i and j are inversion(disorder) if i>j, but i comes before j in this permutation, that is, if the larger number is to the left of the smaller one.

The permutation is called even(or odd), if it has an even (odd) total number of inversions. The operation by which one goes from one permutation to another, composed of the same n numbers, is called substitution nth degree.

A substitution that transforms one permutation into another is written in two lines in common brackets, and the numbers occupying the same places in the permutations under consideration are called relevant and are written one under the other. For example, the symbol represents a substitution in which 3 becomes 4, 1 → 2, 2 → 1, 4 → 3. The substitution is called even(or odd), if the total number of inversions in both substitution strings is even (odd). Any substitution of the nth degree can be written in the form, i.e. with natural numbers in the top line.

Let us be given a square matrix of order n

Let us consider all possible products of n elements of this matrix, taken one and only one from each row and each column, i.e. works of the form:

, (4.4)

where the indices q 1, q 2,...,q n constitute some permutation of numbers
1, 2,..., n. The number of such products is equal to the number of different permutations of n symbols, i.e. equals n!. The sign of the product (4.4) is equal to (- 1) q, where q is the number of inversions in the permutation of the second indices of the elements.

Determinant The nth order corresponding to matrix (4.3) is called the algebraic sum n! members of the form (4.4). To write a determinant, use the symbol or detA = (determinant, or determinant, of matrix A).

Properties of determinants

1. The determinant does not change during transposition.

2. If one of the lines of the determinant consists of zeros, then the determinant is equal to zero.

3. If two lines in the determinant are rearranged, the determinant will change sign.

4. A determinant containing two identical strings is equal to zero.

5. If all elements of a certain row of the determinant are multiplied by some number k, then the determinant itself will be multiplied by k.

6. A determinant containing two proportional lines is equal to zero.

7. If all elements of the i-th row of the determinant are presented as the sum of two terms a i j = b j + c j (j = 1,...,n), then the determinant is equal to the sum of determinants for which all rows except the i-th are - are the same as in the given determinant, and the i-th row in one of the terms consists of elements b j , in the other - of elements c j .

8. The determinant does not change if the corresponding elements of another row are added to the elements of one of its rows, multiplied by the same number.

Comment. All properties remain valid if we take columns instead of rows.

Minor M i j of the element a i j of the determinant d of the nth order is called the determinant of order n-1, which is obtained from d by deleting the row and column containing this element.

Algebraic complement element a i j of the determinant d is called its minor M i j , taken with the sign (-1) i + j . The algebraic complement of an element a i j will be denoted by A i j . Thus, A i j = (-1) i + j M i j .

Methods for practical calculation of determinants, based on the fact that a determinant of order n can be expressed in terms of determinants of lower orders, is given by the following theorem.

Theorem (decomposition of the determinant in a row or column).

The determinant is equal to the sum of the products of all elements of its arbitrary row (or column) by their algebraic complements. In other words, d is expanded into elements of the i-th row

d = a i 1 A i 1 + a i 2 A i 2 +... + a i n A i n (i = 1,...,n)

or jth column

d = a 1 j A 1 j + a 2 j A 2 j +... + a n j A n j (j =1,...,n).

In particular, if all but one element of a row (or column) is zero, then the determinant is equal to that element multiplied by its algebraic complement.

Formula for calculating the third order determinant.

To make this formula easier to remember:

Example 2.4. Without calculating the determinant, show that it is equal to zero.

Solution. Subtracting the first from the second line, we obtain a determinant equal to the original one. If we also subtract the first from the third line, we get a determinant in which the two lines are proportional. This determinant is equal to zero.

Example 2.5. Calculate the determinant D = by expanding it into the elements of the second column.

Solution. Let's expand the determinant into the elements of the second column:

D = a 12 A 12 + a 22 A 22 +a 32 A 32 =

.

Example 2.6. Compute determinant

,

in which all elements on one side of the main diagonal are equal to zero.

Solution. Let us expand the determinant of A along the first line:

.

The determinant on the right can be expanded again along the first line, then we get:

.

Example 2.7. Compute determinant .

Solution. If you add the first line to each line of the determinant, starting from the second, you will get a determinant in which all elements below the main diagonal will be equal to zero. Namely, we get the determinant: , equal to the original one.

Reasoning as in the previous example, we find that it is equal to the product of the elements of the main diagonal, i.e. n!. The method by which this determinant is calculated is called the method of reduction to triangular form.

It is possible to match some number, calculated according to a certain rule and called determinant.

The need to introduce the concept determinant - numbers, characterizing square order matrix n , is closely related to solving systems of linear algebraic equations.

Matrix determinant A we will denote: | A| or D.

Determinant of a first order matrixA = (A 11) the element is called A eleven . For example, for A= (-4) we have | A| = -4.

Determinant of a second order matrix called number, determined by the formula

|A| = .

For example, | A| = .

In words, this rule can be written as follows: with your sign you need to take the product of elements connected main diagonal, and the product of elements connected by the vertices of triangles for which base parallel to main diagonal. With the opposite sign, similar products are taken, only with respect to the secondary diagonal.

For example,

Definition of the determinant of a matrix n We will not give the order, but will only show the method for finding it.

Later, instead of words matrix determinant n-th order let's just talk determinant n-th order. Let's introduce new concepts.

Let a square matrix be given n-th order.

MinorM element ij A ij matrices A called determinant (n-1)th order obtained from the matrix A by crossing out i-th line and j th column.

The algebraic complement A ij of an element a ij of matrix A is its minor, taken with the sign (-1) i+j:

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

those. the algebraic complement either coincides with its minor when the sum of the row and column numbers is an even number, or differs from it in sign when the sum of the row and column numbers is an odd number.

For example, for elements A 11 and A 12 matrices A = minors

M 11 = A 11 = ,

M 12 = ,

A A 12 = (-1) 1+2 M 12 = -8.

Theorem (about the expansion of the determinant) . The determinant of a square matrix is ​​equal to the sum of the products of the elements of any row (column) by their algebraic complements, i.e.

|A| = A i1 A i1+ A i2 A i2 + … + A in A in,
for anyone i = 1, 2, …, n

|A| = A 1j A 1j + A 2j A 2j + … + A nj A nj,

for anyone j = 1, 2, …, n

The first formula is called i-th line, and the second - expansion of the determinant into elements j th column.

It is easy to understand that with the help of these formulas any determinant n The th order can be reduced to the sum of determinants, the order of which will be 1 less, etc. until we reach determinants of the 3rd or 2nd order, the calculation of which is no longer difficult.

To find the determinant, the following basic properties can be applied:

1. If any row (or column) of the determinant consists of zeros, then the determinant itself is equal to zero.

2. When rearranging any two rows (or two columns), the determinant is multiplied by -1.

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

4. The common factor of the elements of any row (or column) can be taken out of the determinant sign.

5. The value of the determinant will not change if all rows and columns are swapped.

6. The value of the determinant will not change if another row (column) multiplied by any number is added to one of the rows (or to one of the columns).

7. The sum of the products of the elements of any row (or column) of a matrix by the algebraic complements of the elements of another row (column) of this matrix is ​​equal to zero.

8. The determinant of the product of two square matrices is equal to the product of their determinants.

The introduction of the concept of the determinant of a matrix allows us to define another operation with matrices - finding the inverse of a matrix.

For every non-zero number there is an inverse number such that the product of these numbers gives one. For square matrices there is also such a concept.

Matrix A-1 is called reverse towards square matrix A, if when multiplying this matrix by the given one both on the right and on the left it turns out identity matrix, i.e.

A× A -1 = A-1× A= E.

From the definition it follows that only a square matrix has an inverse; in this case, the inverse matrix will be square of the same order. However, not every square matrix has its inverse.

Determinant: det, ||, determinant.

The determinant is not a matrix, but a number.

How to find the determinant of a matrix?

To find the determinant of a matrix, introduce the concept "minor". Designation: M ij - minor, M ij 2 - second order minor (determinant of the 2*2 matrix), etc.

To find the minor for the element a ij , we delete the i-th row and j-th column from the matrix A. We obtain a matrix of dimension n-1*m-1, we find determinant of this matrix.

Example: find the second order minor for element a 12 of matrix A:

We cross out the 1st row and 2nd column from matrix A. We obtain a matrix of dimension 2*2, find determinant of this matrix:

Thus, minor is not a matrix, but a number.

Example: find the determinant (in general form) of a 2*2 matrix by decomposing along 1) rows; 2) column:

By line: det A = a 11 *(-1) 1+1 *M 11 +a 12 *(-1) 1+2 *M 12 = a 11 *1*a 22 +a 12 *(-1)* a 21 =
= a 11 *a 22 -a 12 *a 21

By column: det A = a 11 *(-1) 1+1 *M 11 +a 21 *(-1) 2+1 *M 21 = a 11 *1*a 22 +a 21 *(-1)* a 12 =
= a 11 *a 22 -a 21 *a 12

It is easy to see that the same result is obtained.

Thus, to find the determinant of a 2*2 matrix, it is enough to subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal:

How to quickly calculate the third order determinant?

To calculate the third order determinant, use triangle rule(or "stars").

1. Multiply the elements of the main diagonal: det(A)=11*22*33...

2. To the resulting product we add the product of “triangles with bases parallel to the main diagonal”: det(A)=11*22*33+31*12*23+13*21*32...

3. We take everything connected with the secondary diagonal with a “-” sign. We multiply the elements of the secondary diagonal and subtract: det(A)=11*22*33+31*12*23+13*21*32-13*22*31...

4. Similarly to the “main triangles”, we multiply the side triangles and subtract: det(A)=11*22*33+31*12*23+13*21*32-13*22*31-11*23*32-33*12 *21.

det(A)=11*22*33+31*12*23+13*21*32-13*22*31-11*23*32-33*12*21=
=7986+8556+8736-8866-8096-8316=0

Properties of the determinant of a matrix.

• When two parallel rows or columns of a determinant are swapped, its sign is reversed;
• The determinant containing two identical rows or columns is equal to zero;
• If one of the lines of the determinant is multiplied by any number, the result is a determinant equal to the original determinant multiplied by this number;
• When a matrix is ​​transposed, its determinant does not change its value;
• If in the determinant, instead of any line, we write the sum of this line and any other line, multiplied by a certain number, then the resulting new determinant will be equal to the original one;
• If each element of any row or column of a determinant is represented as a sum of two terms, then this determinant can be decomposed into the sum of two corresponding determinants;
• The common factor of the elements of any row or column of the determinant can be taken out of the sign of the determinant.

· Determinant square matrices A of nth order or determinant of nth order is a number equal to an algebraic sum P! members, each of which is a product P matrix elements taken one from each row and each column with certain signs. The determinant is denoted by or.

Second order determinant is a number expressed as follows: . For example .

Third order determinant calculated using the triangle rule (Sarrus rule): .

Example. .

Comment. In practice, third-order determinants, as well as higher-order ones, are calculated using the properties of determinants.

Properties of nth order determinants.

1. The value of the determinant will not change if each row (column) is replaced by a column (row) with the same number - transpose.

2. If one of the rows (columns) of the determinant consists of zeros, then the value of the determinant is zero.

3. If two rows (columns) are swapped in the determinant, then the absolute value of the determinant will not change, but the sign will change to the opposite.

4. A determinant containing two identical rows (columns) is equal to zero.

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

· Minor some element of the determinant P-th order is called the determinant ( P-1)th order, obtained from the original by crossing out the row and column at the intersection of which the selected element is located. Designation: .

· Algebraic complement the element of the determinant is called its minor, taken with the sign. Designation: T.o. =.

6. The determinant of a square matrix is ​​equal to the sum of the products of the elements of any row (or column) by their algebraic complements ( decomposition theorem).

7. If each element of the -th row represents a sum k terms, then the determinant is represented as a sum k determinants in which all lines except the -th line are the same as in the original determinant, and the -th line in the first determinant consists of the first terms, in the second - of the second, etc. The same is true for columns.

8. The determinant will not change if another row (column) is added to one of the rows (columns), multiplied by the number.

Consequence. If a linear combination of its other rows (columns) is added to a row (column) of a determinant, then the determinant will not change.

9. The determinant of a diagonal matrix is ​​equal to the product of the elements on the main diagonal, i.e.

Comment. The determinant of a triangular matrix is ​​also equal to the product of the elements on the main diagonal.

The listed properties of determinants make it possible to significantly simplify their calculation, which is especially important for determinants of high orders. In this case, it is advisable to transform the original matrix so that the transformed matrix has a row or column containing as many zeros as possible (“zeroing” rows or columns).

Examples. Let's calculate the determinant given in the previous example again, using the properties of determinants.

Solution: Note that in the first line there is a common factor - 2, and in the second - a common factor 3, let's take them out of the determinant sign (by property 5). Next, we expand the determinant, for example, in the first column, using property 6 (expansion theorem).

Most effective method of reducing the determinant to diagonal or triangular form . To calculate the determinant of a matrix, it is enough to perform a matrix transformation that does not change the determinant and allows you to turn the matrix into a diagonal one.

In conclusion, we note that if the determinant of a square matrix is ​​equal to zero, then the matrix is ​​called degenerate (or special) , otherwise - non-degenerate .

Equal to the sum of the products of the elements of a row or column by their algebraic complements, i.e. , where i 0 is fixed.
Expression (*) is called the expansion of the determinant D into elements of the row numbered i 0 .

Purpose of the service. This service is designed to find the determinant of a matrix online with the entire solution process recorded in Word format. Additionally, a solution template is created in Excel.

Instructions. Select the matrix dimension, click Next. The determinant can be calculated in two ways: a-priory And by row or column. If you need to find the determinant by creating zeros in one of the rows or columns, you can use this calculator.

Algorithm for finding the determinant

1. For matrices of order n=2, the determinant is calculated using the formula: Δ=a 11 *a 22 -a 12 *a 21
2. For matrices of order n=3, the determinant is calculated through algebraic additions or Sarrus method.
3. A matrix having a dimension greater than three is decomposed into algebraic complements, for which their determinants (minors) are calculated. For example, 4th order matrix determinant found through expansion into rows or columns (see example).

To calculate the determinant containing functions in a matrix, standard methods are used. For example, calculate the determinant of a 3rd order matrix:

We use the method of decomposition along the first row.
Δ = sin(x)× + 1× = 2sin(x)cos(x)-2cos(x) = sin(2x)-2cos(x)

Methods for calculating determinants

Finding the determinant through algebraic additions is a common method. A simplified version of it is the calculation of the determinant by Sarrus' rule. However, when the matrix dimension is large, the following methods are used:

1. calculating the determinant using the order reduction method
2. calculation of the determinant using the Gaussian method (by reducing the matrix to triangular form).

In Excel, the function =MOPRED(cell range) is used to calculate the determinant.

Applied use of determinants

Determinants are calculated, as a rule, for a specific system specified in the form of a square matrix. Let's consider some types of problems on finding the determinant of a matrix. Sometimes you need to find an unknown parameter a for which the determinant would be equal to zero. To do this, it is necessary to create a determinant equation (for example, according to triangle rule) and, equating it to 0, calculate the parameter a.
column decomposition (first column):
Minor for (1,1): Cross out the first row and first column from the matrix.
Let's find a determinant for this minor. ∆ 1.1 = (2 (-2)-2 1) = -6.

Let's determine the minor for (2,1): to do this, we delete the second row and the first column from the matrix.

Let's find a determinant for this minor. ∆ 2.1 = (0 (-2)-2 (-2)) = 4. Minor for (3,1): Cross out the 3rd row and 1st column from the matrix.
Let's find a determinant for this minor. ∆ 3.1 = (0 1-2 (-2)) = 4
The main determinant is: ∆ = (1 (-6)-3 4+1 4) = -14

Let's find the determinant using row-by-row expansion (by the first row):
Minor for (1,1): Cross out the first row and first column from the matrix.

Let's find a determinant for this minor. ∆ 1.1 = (2 (-2)-2 1) = -6. Minor for (1,2): Cross out the 1st row and 2nd column from the matrix. Let us calculate the determinant for this minor. ∆ 1.2 = (3 (-2)-1 1) = -7. And to find the minor for (1,3), we cross out the first row and third column from the matrix. Let's find a determinant for this minor. ∆ 1.3 = (3 2-1 2) = 4
Find the main determinant: ∆ = (1 (-6)-0 (-7)+(-2 4)) = -14