Often at the university there are tasks in higher mathematics in which it is necessary calculate matrix determinant. By the way, the determinant can only be in square matrices. Below we consider the basic definitions of what properties the determinant has and how to calculate it correctly. We will also show a detailed solution using examples.

What is the determinant of a matrix: calculating the determinant using the definition

Matrix determinant

The second order is the number.

The determinant of a matrix is ​​denoted - (short for Latin name determinant), or .

If: then it turns out

We recall a few more auxiliary definitions:

Definition

An ordered set of numbers that consists of elements is called a permutation order.

For a set that contains elements, there is a factorial (n), which is always denoted by an exclamation point: . Permutations differ from each other only in their order. To make it clearer, let's take an example:

Consider a set of three elements (3, 6, 7). There are 6 permutations in total, since .:

Definition

An inversion in a permutation of the order is an ordered set of numbers (it is also called a bijection), where two of them form a kind of disorder. This is when the larger of the numbers in a given permutation is located to the left of the smaller number.

Above, we considered an example with the inversion of a permutation, where there were numbers. So, let's take the second line, where, judging by the given numbers, it turns out that , and , since the second element is greater than the third element . Let's take the sixth line for comparison, where the numbers are located: . There are three pairs here: , and , since title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: 0px;">; , так как title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: 0px;">; , – title="Rendered by QuickLaTeX.com" height="12" width="43" style="vertical-align: 0px;">.!}

We will not study the inversion itself, but the permutations will be very useful to us in the further consideration of the topic.

Definition

Determinant of matrix x - number:

is a permutation of numbers from 1 to an infinite number , and is the number of inversions in the permutation. Thus, the determinant includes terms, which are called “terms of the determinant”.

You can calculate the determinant of a matrix of the second order, third and even fourth. Also worth mentioning:

Definition

the determinant of a matrix is ​​a number that is equal to

To understand this formula, we will describe it in more detail. The determinant of a square matrix x is a sum that contains terms, and each term is a product of a certain number of matrix elements. At the same time, each product has an element from each row and each column of the matrix.

It may appear in front of a certain term if the elements of the matrix in the product go in order (by row number), and the number of inversions in the permutation of the set of column numbers is odd.

It was mentioned above that the matrix determinant is denoted by or , that is, the determinant is often called the determinant.

So, back to the formula:

It can be seen from the formula that the determinant of a first-order matrix is ​​an element of the same matrix.

Calculation of the determinant of a matrix of the second order

Most often, in practice, the matrix determinant is solved by methods of the second, third, and less often, the fourth order. Consider how the determinant of a second-order matrix is ​​calculated:

In a matrix of the second order , it follows that the factorial . Before applying the formula

It is necessary to determine what data we get:

2. permutations of sets: and ;

3. number of inversions in permutation : and , since title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: -1px;">;!}

4. corresponding works : and .

It turns out:

Based on the above, we get a formula for calculating the determinant of a second-order square matrix, that is, x:

Consider on specific example how to calculate the determinant of a second order square matrix:

Example

Task

Calculate the determinant of matrix x :

Solution

So, we get , , , .

To solve it, you need to use the previously considered formula:

We substitute the numbers from the example and find:

Answer

Second order matrix determinant = .

Calculation of the determinant of a matrix of the third order: an example and a solution using the formula

Definition

The determinant of a third-order matrix is ​​the number obtained from nine given numbers arranged in a square table,

The third order determinant is found in much the same way as the second order determinant. The only difference is in the formula. Therefore, if you are well versed in the formula, then there will be no problems with the solution.

Consider a third-order square matrix * :

Based on this matrix, we understand that, respectively, the factorial = , which means that the total permutations are obtained

To apply the formula correctly, you need to find the data:

So, total permutations of the set :

The number of inversions in the permutation , and the corresponding products = ;

Number of inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="65" style="vertical-align: -4px;">, соответствующие произведения = ;!}

Permutation inversions title="Rendered by QuickLaTeX.com" height="18" width="65" style="vertical-align: -4px;"> ;!}

. ; inverses in permutation title="Rendered by QuickLaTeX.com" height="18" width="118" style="vertical-align: -4px;">, соответствующие произведение = !}

. ; inverses in permutation title="Rendered by QuickLaTeX.com" height="18" width="118" style="vertical-align: -4px;">, соответствующие произведение = !}

. ; inverses in permutation title="Rendered by QuickLaTeX.com" height="18" width="171" style="vertical-align: -4px;">, соответствующие произведение = .!}

Now we get:

Thus, we have obtained a formula for calculating the determinant of a matrix of order x:

Finding a matrix of the third order by the triangle rule (Sarrus rule)

As mentioned above, the elements of the 3rd order determinant are located in three rows and three columns. If you enter the notation of the general element , then the first element denotes the row number, and the second element from the indices, the column number. There are main (elements) and secondary (elements) diagonals of the determinant. The terms on the right side are called terms of the determinant).

It can be seen that each member of the determinant is in the schema with only one element in each row and each column.

You can calculate the determinant using the rectangle rule, which is shown as a diagram. The determinant members from the elements of the main diagonal are highlighted in red, as well as the terms from the elements that are at the vertex of triangles that have one side, are parallel to the main diagonal (left diagram), are taken with the sign.

The terms with blue arrows from the elements of the side diagonal, as well as from the elements that are at the vertices of triangles that have sides parallel to the side diagonal (right diagram) are taken with the sign.

In the following example, we will learn how to calculate the determinant of a third-order square matrix.

Example

Task

Calculate the determinant of the matrix of the third order:

Solution

In this example:

We calculate the determinant using the formula or scheme discussed above:

Answer

Third order matrix determinant =

Basic properties of third-order matrix determinants

Based on the previous definitions and formulas, consider the main matrix determinant properties.

1. The size of the determinant will not change when the corresponding rows, columns are replaced (such a replacement is called a transposition).

Using an example, make sure that the determinant of the matrix is ​​equal to the determinant of the transposed matrix:

Recall the formula for calculating the determinant:

We transpose the matrix:

We calculate the determinant of the transposed matrix:

We made sure that the determinant of the transported matrix is ​​equal to the original matrix, which indicates the correct solution.

2. The sign of the determinant will change to the opposite if any two of its columns or two rows are interchanged in it.

Let's look at an example:

Given two third-order matrices ( x ):

It is necessary to show that the determinants of these matrices are opposite.

Solution

In the matrix and in the matrix rows have changed (the third from the first, and from the first to the third). According to the second property, the determinants of two matrices must differ in sign. That is, one matrix is ​​positive and the other is negative. let's check this property by applying the formula to calculate the determinant.

The property is true because .

3. The determinant is equal to zero if it has the same corresponding elements in two rows (columns). Let the determinant have the same elements of the first and second columns:

Swapping the same columns, we, according to property 2, get a new determinant: = . On the other hand, the new determinant is the same as the original one, since the answers are the same elements, i.e. = . From these equalities we get: = .

4. The determinant is equal to zero if all elements of one row (column) are zeros. This statement emerges from the fact that each term of the determinant according to formula (1) has one, and only one element from each row (column), which has only zeros.

Let's look at an example:

Let us show that the matrix determinant is equal to zero:

Our matrix has two identical columns (second and third), therefore, based on given property, the determinant must be zero. Let's check:

Indeed, the determinant of a matrix with two identical columns is zero.

5. The common factor of the elements of the first row (column) can be taken out of the determinant sign:

6. If the elements of one row or one column of the determinant are proportional to the corresponding elements of the second row (column), then such a determinant is equal to zero.

Indeed, after property 5, the proportionality coefficient can be taken out of the sign of the determinant, and then property 3 can be used.

7. If each of the elements of the rows (columns) of the determinant is the sum of two terms, then this determinant can be given as the sum of the corresponding determinants:

To check, it is enough to write in expanded form according to (1) the determinant that is on the left side of the equality, then separately group the terms that contain elements and . Each of the resulting groups of terms will be the first and second determinants on the right side of the equality, respectively.

8. The values ​​of the definition will not change if the corresponding elements of the second row (column) multiplied by the same number are added to the element of one row or one column:

This equality is obtained from properties 6 and 7.

9. The determinant of the matrix , , is equal to the sum of the products of the elements of any row or column and their algebraic complements.

Here by means the algebraic complement of the matrix element . Using this property, you can calculate not only matrices of the third order, but also matrices of higher orders ( x or x ). In other words, this is a recursive formula that is needed in order to calculate the determinant of a matrix of any order. Remember it, as it is often used in practice.

It is worth saying that using the ninth property, one can calculate the determinants of matrices not only of the fourth order, but also of higher orders. However, in this case, you need to perform a lot of computational operations and be careful, since the slightest mistake in the signs will lead to an incorrect decision. Matrices of higher orders are most conveniently solved by the Gaussian method, and we will talk about this later.

10. The determinant of the product of matrices of the same order is equal to the product of their determinants.

Let's look at an example:

Example

Task

Make sure that the determinant of the two matrices and is equal to the product of their determinants. Given two matrices:

Solution

First, we find the product of the determinants of two matrices and .

Now we perform the multiplication of both matrices and thus, we calculate the determinant:

Answer

We made sure that

Calculating the determinant of a matrix using the Gaussian method

Matrix determinant updated: November 22, 2019 by: Scientific Articles.Ru

Exercise. Calculate the determinant by expanding it over the elements of some row or some column.

Solution. Let us first perform elementary transformations on the rows of the determinant by making as many zeros as possible either in a row or in a column. To do this, first we subtract nine thirds from the first line, five thirds from the second, and three thirds from the fourth, we get:

We expand the resulting determinant by the elements of the first column:

The resulting third-order determinant is also expanded by the elements of the row and column, having previously obtained zeros, for example, in the first column. To do this, we subtract two second lines from the first line, and the second from the third:

Answer.

12. Slough 3 orders

1. Rule of the triangle

Schematically, this rule can be represented as follows:

The product of elements in the first determinant that are connected by lines is taken with a plus sign; similarly, for the second determinant, the corresponding products are taken with a minus sign, i.e.

2. Sarrus rule

To the right of the determinant, the first two columns are added and the products of the elements on the main diagonal and on the diagonals parallel to it are taken with a plus sign; and the products of the elements of the secondary diagonal and the diagonals parallel to it, with a minus sign:

3. Expansion of the determinant in a row or column

The determinant is equal to the sum of the products of the elements of the row of the determinant and their algebraic complements. Usually choose the row/column in which/th there are zeros. The row or column on which the decomposition is carried out will be indicated by an arrow.

Exercise. Expanding over the first row, calculate the determinant

Solution.

Answer.

4. Bringing the determinant to a triangular form

With the help of elementary transformations over rows or columns, the determinant is reduced to a triangular form, and then its value, according to the properties of the determinant, is equal to the product of the elements on the main diagonal.

Example

Exercise. Compute determinant bringing it to a triangular shape.

Solution. First, we make zeros in the first column under the main diagonal. All transformations will be easier to perform if the element is equal to 1. To do this, we will swap the first and second columns of the determinant, which, according to the properties of the determinant, will cause it to change sign to the opposite:

Next, we get zeros in the second column in place of the elements under the main diagonal. And again, if the diagonal element is equal to , then the calculations will be simpler. To do this, we swap the second and third lines (and at the same time change to the opposite sign of the determinant):

Next, we make zeros in the second column under the main diagonal, for this we proceed as follows: we add three second rows to the third row, and two second rows to the fourth, we get:

Further, from the third row we take out (-10) as a determinant and make zeros in the third column under the main diagonal, and for this we add the third to the last row:

In the course of solving problems in higher mathematics, it is very often necessary to calculate matrix determinant. The determinant of a matrix appears in linear algebra, analytical geometry, mathematical analysis and other sections of higher mathematics. Thus, one simply cannot do without the skill of solving determinants. Also, for self-testing, you can download the determinant calculator for free, it will not teach you how to solve determinants by itself, but it is very convenient, because it is always beneficial to know the correct answer in advance!

I will not give a strict mathematical definition of the determinant, and, in general, I will try to minimize mathematical terminology, this will not make it easier for most readers. The purpose of this article is to teach you how to solve second, third and fourth order determinants. All the material is presented in a simple and accessible form, and even a full (empty) kettle in higher mathematics, after careful study of the material, will be able to correctly solve the determinants.

In practice, most often you can find a second-order determinant, for example: , and a third-order determinant, for example: .

Fourth order determinant is also not an antique, and we will come to it at the end of the lesson.

I hope everyone understands the following: The numbers inside the determinant live on their own, and there is no question of any subtraction! You can't swap numbers!

(In particular, it is possible to perform pairwise permutations of rows or columns of a determinant with a change of its sign, but often this is not necessary - see the next lesson Properties of a determinant and lowering its order)

Thus, if any determinant is given, then do not touch anything inside it!

Notation: If given a matrix , then its determinant is denoted by . Also, very often the determinant is denoted by a Latin letter or Greek.

1)What does it mean to solve (find, reveal) a determinant? To calculate the determinant is to FIND THE NUMBER. The question marks in the above examples are completely ordinary numbers.

2) Now it remains to figure out HOW to find this number? To do this, you need to apply certain rules, formulas and algorithms, which will be discussed now.

Let's start with the determinant "two" to "two":

THIS SHOULD BE REMEMBERED, at least for the time of studying higher mathematics at the university.

Let's look at an example right away:

Ready. Most importantly, DO NOT CONFUSE THE SIGNS.

Three-by-three matrix determinant can be opened in 8 ways, 2 of them are simple and 6 are normal.

Let's start with two simple ways

Similar to the “two by two” determinant, the “three by three” determinant can be expanded using the formula:

The formula is long and it is easy to make a mistake due to inattention. How to avoid embarrassing mistakes? For this, a second method for calculating the determinant was invented, which actually coincides with the first. It is called the Sarrus method or the "parallel strips" method.
The bottom line is that the first and second columns are attributed to the right of the determinant and the lines are carefully drawn with a pencil:

Factors located on the "red" diagonals are included in the formula with a "plus" sign.
Factors located on the "blue" diagonals are included in the formula with a minus sign:

Example:

Compare the two solutions. It is easy to see that this is the SAME, just in the second case the factors of the formula are slightly rearranged, and, most importantly, the probability of making a mistake is much less.

Now consider the six normal ways to calculate the determinant

Why normal? Because in the vast majority of cases, determinants need to be opened in this way.

As you can see, the three-by-three determinant has three columns and three rows.
You can solve the determinant by expanding it on any row or on any column.
Thus, it turns out 6 ways, while in all cases using of the same type algorithm.

The matrix determinant is equal to the sum of the products of the row (column) elements and the corresponding algebraic additions. Scary? Everything is much simpler, we will use an unscientific, but understandable approach, accessible even to a person who is far from mathematics.

In the following example, we will expand the determinant on the first line.
To do this, we need a matrix of signs: . It is easy to see that the signs are staggered.

Attention! The matrix of signs is my own invention. This concept not scientific, it does not need to be used in the final design of assignments, it only helps you understand the algorithm for calculating the determinant.

I'll give the complete solution first. Again, we take our experimental determinant and perform calculations:

AND main question: HOW to get this from the “three by three” determinant:
?

So, the “three by three” determinant comes down to solving three small determinants, or as they are also called, MINORS. I recommend remembering the term, especially since it is memorable: minor - small.

As soon as the method of expansion of the determinant is chosen on the first line, obviously everything revolves around it:

Elements are usually viewed from left to right (or top to bottom if a column would be selected)

Let's go, first we deal with the first element of the string, that is, with the unit:

1) We write out the corresponding sign from the matrix of signs:

2) Then we write the element itself:

3) MENTALLY cross out the row and column in which the first element is:

The remaining four numbers form the determinant "two by two", which is called MINOR given element (unit).

We pass to the second element of the line.

4) We write out the corresponding sign from the matrix of signs:

5) Then we write the second element:

6) MENTALLY cross out the row and column containing the second element:

Well, the third element of the first line. No originality

7) We write out the corresponding sign from the matrix of signs:

8) Write down the third element:

9) MENTALLY cross out the row and column in which the third element is:

The remaining four numbers are written in a small determinant.

The rest of the steps are not difficult, since we already know how to count the “two by two” determinants. DO NOT CONFUSE THE SIGNS!

Similarly, the determinant can be expanded over any row or over any column. Naturally, in all six cases the answer is the same.

The determinant "four by four" can be calculated using the same algorithm.
In this case, the matrix of signs will increase:

In the following example, I expanded the determinant on the fourth column:

And how it happened, try to figure it out on your own. More information will come later. If anyone wants to solve the determinant to the end, the correct answer is: 18. For training, it is better to open the determinant in some other column or other line.

To practice, to reveal, to make calculations is very good and useful. But how much time will you spend on a big determinant? Isn't there a faster and more reliable way? I suggest you familiarize yourself with effective methods calculation of determinants in the second lesson - Properties of the determinant. Reducing the order of the determinant .

BE CAREFUL!

It is equal to the sum of the products of the elements of some row or column and their algebraic complements, i.e. , where i 0 is fixed.
The expression (*) is called the decomposition of the determinant D in terms of the elements of the row with the number i 0 .

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

Instruction. Select the dimension of the matrix, click Next. There are two ways to calculate the determinant: a-priory And decomposition by row or column. If you want to find the determinant by creating zeros in one of the rows or columns, then you can use this calculator.

Algorithm for finding the determinant

1. For matrices of order n=2, the determinant is calculated by 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 with a dimension greater than three is decomposed into algebraic additions, for which their determinants (minors) are calculated. For example, 4th order matrix determinant is found through expansion in rows or columns (see example).

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

Let's use the first line expansion.
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. Its simplified version is the calculation of the determinant by the Sarrus rule. However, with a large matrix dimension, the following methods are used:

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

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

Applied use of determinants

The determinants are calculated, as a rule, for a specific system, given in the form of a square matrix. Consider some types of tasks on finding matrix determinant. Sometimes it is required to find an unknown parameter a for which the determinant would be equal to zero. To do this, it is necessary to draw up an equation for the determinant (for example, according to triangle rule) and, equating it to 0 , calculate the parameter a .
decomposition by columns (by the first column):
Minor for (1,1): Delete the first row and the first column from the matrix.
Let's find the determinant for this minor. ∆ 1,1 \u003d (2 (-2) -2 1) \u003d -6.

We define the minor for (2,1): to do this, we delete the second row and the first column from the matrix.

Let's find the determinant for this minor. ∆ 2,1 = (0 (-2)-2 (-2)) = 4 . Minor for (3,1): Delete the 3rd row and 1st column from the matrix.
Let's find the 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 expansion by rows (by the first row):
Minor for (1,1): Delete the first row and the first column from the matrix.

Let's find the determinant for this minor. ∆ 1,1 \u003d (2 (-2) -2 1) \u003d -6. Minor for (1,2): Delete the 1st row and 2nd column from the matrix. Let us calculate the determinant for this minor. ∆ 1,2 \u003d (3 (-2) -1 1) \u003d -7. And to find the minor for (1,3) we cross out the first row and the third column from the matrix. Let's find the determinant for this minor. ∆ 1.3 = (3 2-1 2) = 4
We find the main determinant: ∆ \u003d (1 (-6) -0 (-7) + (-2 4)) \u003d -14

Definition1. 7. Minor element of the determinant is the determinant obtained from the given by deleting the row and column containing the selected element.

Notation: the selected element of the determinant, its minor.

Example. For

Definition1. 8. Algebraic addition element of the determinant is called its minor if the sum of the indices of the given element i + j is an even number, or the opposite of the minor if i + j is odd, i.e.

Consider another way to calculate third-order determinants - the so-called row or column expansion. To do this, we prove the following theorem:

Theorem 1.1. The determinant is equal to the sum of the products of the elements of any of its rows or columns and their algebraic complements, i.e.

where i=1,2,3.

Proof.

We will prove the theorem for the first row of the determinant, since for any other row or column we can carry out similar reasoning and get the same result.

Let's find algebraic additions to the elements of the first row:

Thus, to calculate the determinant, it suffices to find the algebraic additions to the elements of any row or column and calculate the sum of their products by the corresponding elements of the determinant.

Example. Let us calculate the determinant using the expansion in the first column. Note that in this case it is not required to search, since, consequently, we find and Hence,

Higher order determinants.

Definition1. 9. nth order determinant

is the sum of n! members each of which corresponds to one of n! ordered sets obtained by r pairwise permutations of elements from the set 1,2,…,n.

Remark 1. The properties of 3rd order determinants are also valid for nth order determinants.

Remark 2. In practice, high-order determinants are computed using a row or column expansion. This makes it possible to reduce the order of the calculated determinants and ultimately reduce the problem to finding 3rd order determinants.

Example. Calculate the 4th order determinant using the expansion in the 2nd column. To do this, we find:

Hence,

Laplace's theorem- one of the theorems of linear algebra. It is named after the French mathematician Pierre-Simon Laplace (1749 - 1827), who is credited with formulating this theorem in 1772, although a special case of this theorem on the expansion of the determinant in a row (column) was known to Leibniz.

completeness minor is defined as follows:

The following assertion is true.

The number of minors over which the sum is taken in Laplace's theorem is equal to the number of ways to choose columns from , that is, the binomial coefficient .

Since the rows and columns of a matrix are equivalent with respect to the properties of the determinant, Laplace's theorem can also be formulated for the columns of a matrix.

Row (column) decomposition of the determinant (Corollary 1)

A special case of Laplace's theorem is widely known - 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 - square matrix size . Let some row number or column number of the matrix be also given. Then the determinant can be calculated using the following formulas.