Decomposition of a matrix into row elements. Matrix determinant online

Often in universities we come across problems in higher mathematics in which it is necessary calculate the determinant of a matrix. By the way, the determinant can only be in square matrices. Below we will consider the basic definitions, 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

Second order is a number.

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

If:, then it turns out

Let us recall a few more auxiliary definitions:

Definition

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

For a set that contains elements there is a factorial (n), which is always denoted by an exclamation mark: . The permutations differ from each other only in the order in which they appear. To make it clearer, let's give 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 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 number in a given permutation is located to the left of the smaller number.

Above we looked at an example with the inversion of a permutation, where there were numbers . So, let’s take the second line, where judging by these numbers it turns out that , a , since the second element is greater than the third element. Let's take for comparison the sixth line, 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 permutations will be very useful to us in 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 that are called “terms of the determinant”.

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

Definition

The determinant of a matrix is ​​the number that equals

To understand this formula, let us describe it in more detail. The determinant of a square matrix x is a sum that contains terms, and each term is the product of a certain number of matrix elements. Moreover, in each product there is an element from each row and each column of the matrix.

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

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

So, let's return to the formula:

From the formula it is clear that the determinant of a first-order matrix is ​​an element of the same matrix.

Calculation of the determinant of a second-order matrix

Most often in practice, the determinant of a matrix is ​​solved using methods of the second, third, and less often, fourth order. Let's look at how the determinant of a second-order matrix is ​​calculated:

In a second-order matrix, it follows that the factorial is . Before you apply the formula

It is necessary to determine what data we obtain:

2. permutations of sets: and ;

3. number of inversions in the 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 obtain a formula for calculating the determinant of a second-order square matrix, that is, x:

Let's look at a specific example of how to calculate the determinant of a second-order square matrix:

Example

Task

Calculate the determinant of the matrix x:

Solution

So, we get , , , .

To solve, you need to use the previously discussed formula:

We substitute the numbers from the example and find:

Answer

Second order matrix determinant = .

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

Definition

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

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

Consider a third-order square matrix *:

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

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

So, the total permutations of the set are:

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 = ;!}

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

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

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

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

Now we get:

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

Finding a third-order matrix using 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 designation of the general element, then the first element denotes the row number, and the second element from the indices denotes the column number. There is a 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 term of the determinant is in the diagram with only one element in each row and each column.

You can calculate the determinant using the rectangle rule, which is depicted in the form of a diagram. The terms of the determinant 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 parallel to the main diagonal (left diagram), taken with the sign .

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

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

Example

Task

Calculate the determinant of a third-order matrix:

Solution

In this example:

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

Answer

Determinant of a third-order matrix =

Basic properties of determinants of a third-order matrix

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

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

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

Let us recall the formula for calculating the determinant:

Transpose the matrix:

We calculate the determinant of the transposed matrix:

We have verified 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 swapped.

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

The rows in the matrix and in the matrix 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 has a positive sign, and the second one has a negative sign. Let's check this property by using the formula to calculate the determinant.

The property is true because .

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

By swapping identical columns, we, according to Property 2, obtain a new determinant: = . On the other hand, the new determinant coincides with the original one, since the elements have the same answers, that is, = . 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 determinant of the matrix is ​​equal to zero:

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

Indeed, the determinant of a matrix with two identical columns is equal to 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 a determinant are proportional to the corresponding elements of the second row (column), then such a determinant is equal to zero.

Indeed, following property 5, the coefficient of proportionality 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 presented 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 the elements and . Each of the resulting groups of terms will be, respectively, the first and second determinant on the right side of the equality.

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

This equality is obtained based on 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 a matrix element. Using this property, you can calculate not only third-order matrices, but also matrices of higher orders (x or x). In other words, this is a recurrent 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 it is possible to 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 error in the signs will lead to an incorrect decision. It is most convenient to solve matrices of higher orders using 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 two matrices and is equal to the product of their determinants. Two matrices are given:

Solution

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

Now let's multiply both matrices and thus 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

Further properties are related to the concepts of minor and algebraic complement

Minor element is called a determinant, composed of elements remaining after crossing out the row and column at the intersection of which this element is located. The minor element of the order determinant has order . We will denote it by .

Example 1. Let , Then .

This minor is obtained from A by crossing out the second row and third column.

Algebraic complement element is called the corresponding minor multiplied by , i.e. , where is the number of the row and column at the intersection of which this element is located.

VIII.(Decomposition of the determinant into elements of a certain string). The determinant is equal to the sum of the products of the elements of a certain row and their corresponding algebraic complements.

Example 2. Let , Then

Example 3. Let's find the determinant of the matrix , decomposing it into the elements of the first row.

Formally, this theorem and other properties of determinants are applicable only for determinants of matrices of no higher than third order, since we have not considered other determinants. The following definition will allow us to extend these properties to determinants of any order.

Determinant of the matrix order is a number calculated by sequential application of the expansion theorem and other properties of determinants.

You can check that the result of the calculations does not depend on the order in which the above properties are applied and for which rows and columns. Using this definition, the determinant is uniquely found.

Although this definition does not contain an explicit formula for finding the determinant, it allows one to find it by reducing it to the determinants of matrices of lower order. Such definitions are called recurrent.

Example 4. Calculate the determinant:

Although the factorization theorem can be applied to any row or column of a given matrix, fewer computations are obtained by factoring along the column that contains as many zeros as possible.

Since the matrix does not have zero elements, we obtain them using the property VII. Multiply the first line sequentially by numbers and add it to the lines and get:

Let's expand the resulting determinant along the first column and get:

since the determinant contains two proportional columns.

Some types of matrices and their determinants

A square matrix that has zero elements below or above the main diagonal () is called triangular.

Their schematic structure accordingly looks like: or

.

Matrices are used in mathematics to compactly write systems of linear algebraic or differential equations. In this case, the number of rows of the matrix corresponds to the number of equations, and the number of columns corresponds to the number of unknowns. As a result, solving systems of linear equations is reduced to operations on matrices.

The matrix is ​​written as a rectangular table of elements of a ring or field (for example, integers, complex or real numbers). It is a collection of rows and columns at the intersection of which its elements are located. The size of the matrix is ​​determined by the number of rows and columns.

An important value of any matrix is ​​its determinant, which is calculated using a certain formula. It is necessary to manually perform a number of operations on the matrix in order to calculate its determinant. The determinant can be either positive or negative or equal to zero. To check your calculations of the determinant of the matrix, you can use our online calculator. The online calculator will instantly calculate the determinant of the matrix and give the exact value.

The determinant of a matrix is ​​a kind of characteristic of a matrix, or more precisely, it can be used to determine whether the corresponding system of equations has a solution. The determinant of a matrix is ​​widely used in science, such as physics, with the help of which the physical meaning of many quantities is calculated.

Solving systems of linear algebraic equations

Also, using our calculator you can solve a system of linear algebraic equations (SLAE).

Solving systems of linear algebraic equations is one of the common linear algebra problems. SLAEs and methods for their solution underlie many applied areas, including econometrics and linear programming.

Free online calculator

Our free solver will allow you to solve equations online of any complexity in a matter of seconds. All you need to do is simply enter your data into the calculator. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.

The concept of determinant is one of the main ones in the course of linear algebra. This concept is inherent ONLY to SQUARE MATRICES, and this article is devoted to this concept. Here we will talk about determinants of matrices whose elements are real (or complex) numbers. In this case, the determinant is a real (or complex) number. All further presentation will be an answer to the questions of how to calculate the determinant and what properties it has.

First, we give the definition of the determinant of a square matrix of order n by n as the sum of products of permutations of matrix elements. Based on this definition, we will write down formulas for calculating the determinants of matrices of the first, second, and third orders and analyze in detail the solutions of several examples.

Next, we move on to the properties of the determinant, which we will formulate in the form of theorems without proof. Here we will obtain a method for calculating the determinant through its expansion into the elements of a row or column. This method allows you to reduce the calculation of the determinant of a matrix of order n by n to the calculation of the determinants of matrices of order 3 by 3 or less. We will definitely show solutions to several examples.

In conclusion, we will focus on calculating the determinant using the Gaussian method. This method is good for finding the values ​​of determinants of matrices of order higher than 3 by 3, since it requires less computational effort. We will also look at the solutions to the examples.

Page navigation.

Determination of the determinant of a matrix, calculation of the determinant of a matrix by definition.

Let us recall a few auxiliary concepts.

Definition.

Permutation of order n An ordered set of numbers consisting of n elements is called.

For a set containing n elements, there are n! (n factorial) permutations of order n. Permutations differ from each other only in the order in which the elements appear.

For example, consider a set consisting of three numbers: . Let us write down all the permutations (there are six in total, since ):

Definition.

By inversion in a permutation of order n Any pair of indices p and q for which the p-th element of the permutation is greater than the q-th is called.

In the previous example, the inverse of the permutation 4, 9, 7 is the pair p=2, q=3, since the second element of the permutation is equal to 9 and it is greater than the third, equal to 7. The inversion of the permutation 9, 7, 4 will be three pairs: p=1, q=2 (9>7); p=1, q=3 (9>4) and p=2, q=3 (7>4).

We will be more interested in the number of inversions in the permutation, rather than the inversion itself.

Let be a square matrix of order n by n over the field of real (or complex) numbers. Let be the set of all permutations of order n of the set . The set contains n! permutations. Let us denote the k-th permutation of the set as , and the number of inversions in the k-th permutation as .

Definition.

Matrix determinant And there is a number equal to .

Let's describe this formula in words. The determinant of a square matrix of order n by n is the sum containing n! terms. Each term is a product of n elements of the matrix, and each product contains an element from each row and from each column of the matrix A. A coefficient (-1) appears before the k-th term if the elements of matrix A in the product are ordered by row number, and the number of inversions in the k-th permutation of the set of column numbers is odd.

The determinant of matrix A is usually denoted as , and det(A) is also used. You may also hear the determinant called a determinant.

So, .

From this it is clear that the determinant of a first-order matrix is ​​the element of this matrix.

Calculating the determinant of a second order square matrix - formula and example.

about 2 by 2 in general.

In this case n=2 , therefore n!=2!=2 .

.

We have

Thus, we have obtained a formula for calculating the determinant of a matrix of order 2 by 2, it has the form .

Example.

order .

Solution.

In our example. We apply the resulting formula :

Calculating the determinant of a third order square matrix - formula and example.

Let's find the determinant of a square matrix about 3 by 3 in general.

In this case n=3, therefore n!=3!=6.

Let's arrange in the form of a table the necessary data to apply the formula .

We have

Thus, we have obtained a formula for calculating the determinant of a matrix of order 3 by 3, it has the form

Similarly, you can obtain formulas for calculating the determinants of matrices of order 4 by 4, 5 by 5 and higher. They will look very bulky.

Example.

Calculate the determinant of a square matrix about 3 by 3.

Solution.

In our example

We apply the resulting formula to calculate the determinant of a third-order matrix:

Formulas for calculating the determinants of square matrices of the second and third orders are very often used, so we recommend that you remember them.

Properties of the determinant of a matrix, calculating the determinant of a matrix using properties.

Based on the stated definition, the following are true: properties of the matrix determinant.

The determinant of the matrix A is equal to the determinant of the transposed matrix A T, that is, .

Example.

Make sure the determinant of the matrix is equal to the determinant of the transposed matrix.

Solution.

Let's use the formula to calculate the determinant of a matrix of order 3 by 3:



Transpose matrix A:


Let's calculate the determinant of the transposed matrix:



Indeed, the determinant of the transposed matrix is ​​equal to the determinant of the original matrix.

If in a square matrix all elements of at least one of the rows (one of the columns) are zero, the determinant of such a matrix is ​​equal to zero.

Example.

Check that the determinant of the matrix order 3 by 3 is zero.

Solution.




Indeed, the determinant of a matrix with a zero column is equal to zero.

If you rearrange any two rows (columns) in a square matrix, then the determinant of the resulting matrix will be opposite to the original one (that is, the sign will change).

Example.

Given two square matrices of order 3 by 3 And . Show that their determinants are opposite.

Solution.

Matrix B is obtained from matrix A by replacing the third row with the first, and the first with the third. According to the property considered, the determinants of such matrices must differ in sign. Let's check this by calculating the determinants using the well-known formula.



Really, .

If in a square matrix at least two rows (two columns) are the same, then its determinant is equal to zero.

Example.

Show that the determinant of the matrix equal to zero.

Solution.

In this matrix, the second and third columns are the same, so according to the property considered, its determinant must be equal to zero. Let's check it out.



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

If in a square matrix all the elements of any row (column) are multiplied by a certain number k, then the determinant of the resulting matrix will be equal to the determinant of the original matrix multiplied by k. For example,



Example.

Prove that the determinant of the matrix equal to triple the determinant of the matrix .

Solution.

The elements of the first column of matrix B are obtained from the corresponding elements of the first column of matrix A by multiplying by 3. Then, due to the property considered, the equality must hold. Let's check this by calculating the determinants of matrices A and B.



Therefore, that is what needed to be proven.

PLEASE NOTE.

Do not confuse or mix the concepts of matrix and determinant! The considered property of the determinant of a matrix and the operation of multiplying a matrix by a number are far from the same thing.
, But .

If all the elements of any row (column) of a square matrix represent the sum of s terms (s is a natural number greater than one), then the determinant of such a matrix will be equal to the sum of s determinants of matrices obtained from the original one, if the elements of the row (column) are: leave one term at a time. For example,


Example.

Prove that the determinant of a matrix is ​​equal to the sum of the determinants of matrices .

Solution.

In our example , therefore, due to the considered property of the determinant of the matrix, the equality must be satisfied . Let's check it by calculating the corresponding determinants of matrices of order 2 by 2 using the formula .


From the results obtained it is clear that . This completes the proof.

If the corresponding elements of another row (column) are added to the elements of a certain row (column) of a matrix, multiplied by an arbitrary number k, then the determinant of the resulting matrix will be equal to the determinant of the original matrix.

Example.

Make sure that if to the elements of the third column of the matrix add the corresponding elements of the second column of this matrix, multiplied by (-2), and add the corresponding elements of the first column of the matrix, multiplied by an arbitrary real number, then the determinant of the resulting matrix will be equal to the determinant of the original matrix.

Solution.

If we start from the considered property of the determinant, then the determinant of the matrix obtained after all the transformations specified in the problem will be equal to the determinant of the matrix A.

First, let's calculate the determinant of the original matrix A:



Now let's perform the necessary transformations of matrix A.

Let's add to the elements of the third column of the matrix the corresponding elements of the second column of the matrix, having previously multiplied them by (-2). After this, the matrix will take the form:


To the elements of the third column of the resulting matrix we add the corresponding elements of the first column, multiplied by:


Let's calculate the determinant of the resulting matrix and make sure that it is equal to the determinant of matrix A, that is, -24:



The determinant of a square matrix is ​​equal to the sum of the products of the elements of any row (column) by their algebraic additions.



Here is the algebraic complement of the matrix element , .

This property allows one to calculate the determinants of matrices of order higher than 3 by 3 by reducing them to the sum of several determinants of matrices of order one lower. In other words, this is a recurrent formula for calculating the determinant of a square matrix of any order. We recommend that you remember it due to its fairly frequent applicability.

Let's look at a few examples.

Example.

about 4 by 4, expanding it

• by elements of the 3rd line,
• by elements of the 2nd column.

Solution.

We use the formula for decomposing the determinant into the elements of the 3rd row



We have



So the problem of finding the determinant of a matrix of order 4 by 4 was reduced to calculating three determinants of matrices of order 3 by 3:



Substituting the obtained values, we arrive at the result:


We use the formula for decomposing the determinant into the elements of the 2nd column


and we act in the same way.

We will not describe in detail the calculation of determinants of third-order matrices.



Example.

Compute determinant of matrix about 4 by 4.

Solution.

You can expand the determinant of a matrix into the elements of any column or any row, but it is more profitable to choose the row or column that contains the largest number of zero elements, as this will help avoid unnecessary calculations. Let's expand the determinant into the elements of the first line:



Let us calculate the resulting determinants of matrices of order 3 by 3 using the formula known to us:



Substitute the results and get the desired value


Example.

Compute determinant of matrix about 5 by 5.

Solution.

The fourth row of the matrix has the largest number of zero elements among all rows and columns, so it is advisable to expand the determinant of the matrix precisely according to the elements of the fourth row, since in this case we will need fewer calculations.



The resulting determinants of matrices of order 4 by 4 were found in previous examples, so let’s use the ready-made results:



Example.

Compute determinant of matrix about 7 by 7.

Solution.

You should not immediately rush to sort the determinant into the elements of any row or column. If you look closely at the matrix, you will notice that the elements of the sixth row of the matrix can be obtained by multiplying the corresponding elements of the second row by two. That is, if the corresponding elements of the second row are added to the elements of the sixth row, multiplied by (-2), then the determinant will not change due to the seventh property, and the sixth row of the resulting matrix will consist of zeros. The determinant of such a matrix is ​​equal to zero by the second property.

Answer:

It should be noted that the considered property allows one to calculate the determinants of matrices of any order, but one has to perform a lot of computational operations. In most cases, it is more advantageous to find the determinant of matrices of order higher than the third using the Gaussian method, which we will consider below.

The sum of the products of the elements of any row (column) of a square matrix by the algebraic complements of the corresponding elements of another row (column) is equal to zero.


Example.

Show that the sum of the products of the elements of the third column of the matrix on the algebraic complements of the corresponding elements of the first column is equal to zero.

Solution.




The determinant of the product of square matrices of the same order is equal to the product of their determinants, that is, , where m is a natural number greater than one, A k, k=1,2,...,m are square matrices of the same order.

Example.

Verify that the determinant of the product of two matrices and is equal to the product of their determinants.

Solution.

Let us first find the product of determinants of matrices A and B:


Now let’s perform matrix multiplication and calculate the determinant of the resulting matrix:



Thus, , which is what needed to be shown.

Calculation of the determinant of a matrix using the Gaussian method.

Let us describe the essence of this method. Using elementary transformations, matrix A is reduced to such a form that in the first column all elements except those become zero (this can always be done if the determinant of matrix A is different from zero). We will describe this procedure a little later, but now we will explain why this is done. Zero elements are obtained in order to obtain the simplest expansion of the determinant over the elements of the first column. After such a transformation of the matrix A, taking into account the eighth property and, we obtain

Where - minor (n-1)th order, obtained from matrix A by deleting the elements of its first row and first column.

With the matrix to which minor corresponds, the same procedure is performed to obtain zero elements in the first column. And so on until the final calculation of the determinant.

Now it remains to answer the question: “How to get zero elements in the first column”?

Let us describe the algorithm of actions.

If , then the corresponding elements of the kth row are added to the elements of the first row of the matrix, in which . (If all the elements of the first column of matrix A, without exception, are zero, then its determinant is equal to zero by the second property and no Gaussian method is needed). After such a transformation, the “new” element will be non-zero. The determinant of the “new” matrix will be equal to the determinant of the original matrix due to the seventh property.

Now we have a matrix with . When to the elements of the second line we add the corresponding elements of the first line, multiplied by , to the elements of the third line - the corresponding elements of the first line, multiplied by . And so on. Finally, to the elements of the nth row we add the corresponding elements of the first row, multiplied by . This will result in a transformed matrix A, all elements of the first column of which, except , will be zero. The determinant of the resulting matrix will be equal to the determinant of the original matrix due to the seventh property.

Let's look at the method when solving an example, it will be clearer.

Example.

Calculate the determinant of a matrix of order 5 by 5 .

Solution.

Let's use the Gaussian method. Let's transform matrix A so that all elements of its first column, except , become zero.

Since the element is initially , we add to the elements of the first row of the matrix the corresponding elements, for example, of the second row, since :

The "~" sign indicates equivalence.

Now we add to the elements of the second line the corresponding elements of the first line, multiplied by , to the elements of the third line – the corresponding elements of the first line, multiplied by , and proceed similarly up to the sixth line:

We get

With matrix We carry out the same procedure for obtaining zero elements in the first column:

Hence,

Now we perform transformations with the matrix :

Comment.

At some stage of the matrix transformation using the Gaussian method, a situation may arise when all the elements of the last few rows of the matrix become zero. This will indicate that the determinant is equal to zero.

Let's summarize.

The determinant of a square matrix whose elements are numbers is a number. We looked at three ways to calculate the determinant:

1. through the sum of products of combinations of matrix elements;
2. through the decomposition of the determinant into the elements of a row or column of the matrix;
3. by reducing the matrix to an upper triangular one (Gaussian method).

Formulas were obtained for calculating the determinants of matrices of order 2 by 2 and 3 by 3.

We have examined the properties of the determinant of a matrix. Some of them allow you to quickly understand that the determinant is zero.

When calculating the determinants of matrices of order higher than 3 by 3, it is advisable to use the Gaussian method: perform elementary transformations of the matrix and reduce it to an upper triangular one. The determinant of such a matrix is ​​equal to the product of all elements on the main diagonal.

Let us recall Laplace's theorem:
Laplace's theorem:

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

To calculate determinants, in the general case, k is taken equal to 1. That is, in the determinant d of order n, a row (or column) is arbitrarily chosen. Then the sum of the products of all elements contained in the selected row (or column) and their algebraic complements is equal to the determinant d.

Example:
Compute determinant

Solution:

Let's select an arbitrary row or column. For a reason that will become obvious a little later, we will limit our choice to either the third row or the fourth column. And let's stop on the third line.

Let's use Laplace's theorem.

The first element of the selected row is 10, it appears in the third row and first column. Let us calculate the algebraic complement to it, i.e. Let's find the determinant obtained by crossing out the column and row on which this element stands (10) and find out the sign.

“plus if the sum of the numbers of all rows and columns in which the minor M is located is even, and minus if this sum is odd.”
And we took the minor, consisting of one single element 10, which is in the first column of the third row.

So:


The fourth term of this sum is 0, which is why it is worth choosing rows or columns with the maximum number of zero elements.

Answer: -1228

Example:
Calculate the determinant:

Solution:
Let's select the first column, because... two elements in it are equal to 0. Let us expand the determinant along the first column.


We expand each of the third-order determinants along the first second row


We expand each of the second-order determinants along the first column


Answer: 48
Comment: when solving this problem, formulas for calculating determinants of the 2nd and 3rd orders were not used. Only row or column decomposition was used. Which leads to a decrease in the order of determinants.