Calculate the determinant over the elements of the third row. Reducing the order of the determinant. Expansion of the determinant in a row (column)

For the determinant of the fourth and higher orders, other methods of calculation are usually used than the use of ready-made formulas as for calculating the determinants of the second and third orders. One of the methods for calculating determinants of higher orders is to use the corollary from Laplace's theorem (the theorem itself can be found, for example, in the book by A.G. Kurosh "Course of Higher Algebra"). This corollary allows us to expand the determinant over the elements of some row or column. In this case, the calculation of the determinant of the nth order is reduced to the calculation of n determinants of the (n-1)th order. That is why such a transformation is called lowering the order of the determinant. For example, the calculation of a fourth order determinant is reduced to finding four third order determinants.

Suppose we are given a square matrix of the nth order, i.e. $A=\left(\begin(array) (cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(n1) & a_(n2) & \ldots & a_(nn) \\ \end(array) \right)$. You can calculate the determinant of this matrix by expanding it by row or by column.

Let's fix some string, the number of which is equal to $i$. Then the determinant of the matrix $A_(n\times n)$ can be expanded in the chosen i-th row using the following formula:

\begin(equation) \Delta A=\sum\limits_(j=1)^(n)a_(ij)A_(ij)=a_(i1)A_(i1)+a_(i2)A_(i2)+\ ldots+a_(in)A_(in) \end(equation)

$A_(ij)$ denotes algebraic addition element $a_(ij)$. For detailed information about this concept, I recommend to look at the topic Algebraic additions and minors. The notation $a_(ij)$ denotes an element of a matrix or determinant located at the intersection i-th line j-th column. For more complete information You can look at the theme of the Matrix. Types of matrices. Basic terms.

Let's say we want to find the sum $1^2+2^2+3^2+4^2+5^2$. What phrase can characterize the record $1^2+2^2+3^2+4^2+5^2$? We can say this: this is the sum of one squared, two squared, three squared, four squared and five squared. And you can say it shorter: it is the sum of the squares of integers from 1 to 5. To express the sum more briefly, the notation using the letter $\sum$ (this is the Greek letter "sigma") is used.

Instead of $1^2+2^2+3^2+4^2+5^2$ we can use this notation: $\sum\limits_(i=1)^(5)i^2$. The letter $i$ is called summation index, and the numbers 1 (initial value $i$) and 5 (final value $i$) are called lower and upper summation limits respectively.

Let's decipher the entry $\sum\limits_(i=1)^(5)i^2$ in detail. If $i=1$, then $i^2=1^2$, so the first term of this sum is the number $1^2$:

$$ \sum\limits_(i=1)^(5)i^2=1^2+\ldots $$

The next integer after one is two, so substituting $i=2$, we get: $i^2=2^2$. The amount will now be:

$$ \sum\limits_(i=1)^(5)i^2=1^2+2^2+\ldots $$

After two, the next number is three, so substituting $i=3$ we get: $i^2=3^2$. And the sum will look like:

$$ \sum\limits_(i=1)^(5)i^2=1^2+2^2+3^2+\ldots $$

It remains to substitute only two numbers: 4 and 5. If we substitute $i=4$, then $i^2=4^2$, and if we substitute $i=5$, then $i^2=5^2$. The values ​​of $i$ have reached the upper summation limit, so $5^2$ will be the last term. So the final sum is now:

$$ \sum\limits_(i=1)^(5)i^2=1^2+2^2+3^2+4^2+5^2. $$

This amount can also be calculated by simply adding up the numbers: $\sum\limits_(i=1)^(5)i^2=55$.

For practice, try writing down and calculating the following sum: $\sum\limits_(k=3)^(8)(5k+2)$. The summation index here is the letter $k$, the lower summation limit is 3, and the upper summation limit is 8.

$$ \sum\limits_(k=3)^(8)(5k+2)=17+22+27+32+37+42=177. $$

An analogue of formula (1) also exists for columns. The formula for expanding the determinant in the j-th column is as follows:

\begin(equation) \Delta A=\sum\limits_(i=1)^(n)a_(ij)A_(ij)=a_(1j)A_(1j)+a_(2j)A_(2j)+\ ldots+a_(nj)A_(nj) \end(equation)

The rules expressed by formulas (1) and (2) can be formulated as follows: the determinant is equal to the sum of the products of the elements of a certain row or column and the algebraic complements of these elements. For clarity, consider the fourth-order determinant, written in general form:

$$\Delta=\left| \begin(array) (cccc) a_(11) & a_(12) & a_(13) & a_(14) \\ a_(21) & a_(22) & a_(23) & a_(24) \\ a_(31) & a_(32) & a_(33) & a_(34) \\ a_(41) & a_(42) & a_(43) & a_(44) \\ \end(array) \right| $$

We choose an arbitrary column in this determinant. Take, for example, column number 4. Let's write a formula for expanding the determinant in the chosen fourth column:

Similarly, choosing, for example, the third row, we get the decomposition on this row:

Example #1

Calculate determinant of matrix $A=\left(\begin(array) (ccc) 5 & -4 & 3 \\ 7 & 2 & -1 \\ 9 & 0 & 4 \end(array) \right)$ using expansion on the first row and second column.

We need to calculate the third order determinant $\Delta A=\left| \begin(array) (ccc) 5 & -4 & 3 \\ 7 & 2 & -1 \\ 9 & 0 & 4 \end(array) \right|$. To expand it along the first line, you need to use the formula. We write this expansion in general form:

$$ \Delta A= a_(11)\cdot A_(11)+a_(12)\cdot A_(12)+a_(13)\cdot A_(13). $$

For our matrix $a_(11)=5$, $a_(12)=-4$, $a_(13)=3$. To calculate the algebraic additions $A_(11)$, $A_(12)$, $A_(13)$, we will use formula No. 1 from the topic dedicated to . So, the desired algebraic additions are as follows:

\begin(aligned) & A_(11)=(-1)^2\cdot \left| \begin(array) (cc) 2 & -1 \\ 0 & 4 \end(array) \right|=2\cdot 4-(-1)\cdot 0=8;\\ & A_(12)=( -1)^3\cdot \left| \begin(array) (cc) 7 & -1 \\ 9 & 4 \end(array) \right|=-(7\cdot 4-(-1)\cdot 9)=-37;\\ & A_( 13)=(-1)^4\cdot \left| \begin(array) (cc) 7 & 2 \\ 9 & 0 \end(array) \right|=7\cdot 0-2\cdot 9=-18. \end(aligned)

How did we find algebraic additions? show/hide

Substituting all the found values ​​into the above formula, we get:

$$ \Delta A= a_(11)\cdot A_(11)+a_(12)\cdot A_(12)+a_(13)\cdot A_(13)=5\cdot(8)+(-4) \cdot(-37)+3\cdot(-18)=134. $$

As you can see, we reduced the process of finding a third-order determinant to calculating the values ​​of three second-order determinants. In other words, we lowered the order of the original determinant.

Usually, in such simple cases, the solution is not described in detail, separately finding algebraic additions, and only then substituting them into the formula for calculating the determinant. Most often, they simply continue to write the general formula, until an answer is received. This is how we will decompose the determinant in the second column.

So, let's proceed to the expansion of the determinant in the second column. We will not perform auxiliary calculations, we will simply continue the formula until we get an answer. Note that in the second column, one element is zero, i.e. $a_(32)=0$. This means that the term $a_(32)\cdot A_(32)=0\cdot A_(23)=0$. Using the formula for expanding on the second column, we get:

$$ \Delta A= a_(12)\cdot A_(12)+a_(22)\cdot A_(22)+a_(32)\cdot A_(32)=-4\cdot (-1)\cdot \ left| \begin(array) (cc) 7 & -1 \\ 9 & 4 \end(array) \right|+2\cdot \left| \begin(array) (cc) 5 & 3 \\ 9 & 4 \end(array) \right|=4\cdot 37+2\cdot (-7)=134. $$

Answer received. Naturally, the result of the expansion in the second column coincided with the result of the expansion in the first row, because we were decomposing the same determinant. Note that when expanding on the second column, we did less calculations, since one element of the second column was equal to zero. It is on the basis of such considerations for decomposition that they try to choose the column or row that contains more zeros.

Answer: $\Delta A=134$.

Example #2

Compute matrix determinant $A=\left(\begin(array) (cccc) -1 & 3 & 2 & -3\\ 4 & -2 & 5 & 1\\ -5 & 0 & -4 & 0\\ 9 & 7 & 8 & -7 \end(array) \right)$ using expansion on the selected row or column.

For decomposition, it is most advantageous to choose the row or column that contains the most zeros. Naturally, in this case it makes sense to expand on the third row, since it contains two elements equal to zero. Using the formula, we write the expansion of the determinant in the third row:

$$ \Delta A= a_(31)\cdot A_(31)+a_(32)\cdot A_(32)+a_(33)\cdot A_(33)+a_(34)\cdot A_(34). $$

Since $a_(31)=-5$, $a_(32)=0$, $a_(33)=-4$, $a_(34)=0$, the formula written above becomes:

$$ \Delta A= -5 \cdot A_(31)-4\cdot A_(33). $$

Let us turn to the algebraic complements $A_(31)$ and $A_(33)$. To calculate them, we will use formula No. 2 from the topic on second and third order determinants (in the same section there is detailed examples application of this formula).

\begin(aligned) & A_(31)=(-1)^4\cdot \left| \begin(array) (ccc) 3 & 2 & -3 \\ -2 & 5 & 1 \\ 7 & 8 & -7 \end(array) \right|=10;\\ & A_(33)=( -1)^6\cdot \left| \begin(array) (ccc) -1 & 3 & -3 \\ 4 & -2 & 1 \\ 9 & 7 & -7 \end(array) \right|=-34. \end(aligned)

Substituting the data obtained into the formula for the determinant, we will have:

$$ \Delta A= -5 \cdot A_(31)-4\cdot A_(33)=-5\cdot 10-4\cdot (-34)=86. $$

In principle, the entire solution can be written in one line. If you skip all the explanations and intermediate calculations, then the solution will be written as follows:

$$ \Delta A= a_(31)\cdot A_(31)+a_(32)\cdot A_(32)+a_(33)\cdot A_(33)+a_(34)\cdot A_(34)= \\= -5 \cdot (-1)^4\cdot \left| \begin(array) (ccc) 3 & 2 & -3 \\ -2 & 5 & 1 \\ 7 & 8 & -7 \end(array) \right|-4\cdot (-1)^6\cdot \left| \begin(array) (ccc) -1 & 3 & -3 \\ 4 & -2 & 1 \\ 9 & 7 & -7 \end(array) \right|=-5\cdot 10-4\cdot ( -34)=86. $$

Answer: $\Delta A=86$.

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 matrix determinant in online mode with the execution of the entire course of the decision in Word format. Additionally, a solution template is created in Excel.

Instruction. Select the dimension of the matrix, click Next.

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).

Methods for calculating determinants

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).

Applied use of determinants

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 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.

The second order is a number equal to the difference between the product of the numbers forming the main diagonal and the product of the numbers on the side diagonal, you can find the following designations of the determinant: ; ; ; detA(determinant).

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Example:
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The determinant of a matrix of the third order a number or a mathematical expression is called, calculated according to the following rule

The simplest way to calculate the third-order determinant is to add the determinant of the first two rows from below.

In the formed table of numbers, the elements on the main diagonal and on the diagonals parallel to the main one are multiplied, the sign of the result of the product does not change. next step calculations is a similar multiplication of elements standing on the secondary diagonal and on parallel to it. The signs of the product results are reversed. Then add the resulting six terms.

Example:

Decomposition of the determinant by the elements of some row (column).

Minor M ij element and ij square matrix A called the determinant, composed of the elements of the matrix A, remaining after deletion i- oh line and j-th column.

For example, a minor to an element a 21 third order matrices
there will be a determinant
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We will say that the element and ij occupies an even position if i+j(the sum of the row and column numbers at the intersection of which is given element) - even number, odd place, if i+j- odd number.

Algebraic addition And ij element and ij square matrix A called expression (or the value of the corresponding minor, taken with the “+” sign if the matrix element occupies an even place, and with the “-” sign if the element occupies an odd place).

Example:

a 23= 4;

- algebraic complement of an element a 22= 1.

Laplace's theorem. The determinant is equal to the sum of the products of the elements of some row (column) and their corresponding algebraic additions.

Let's illustrate with the example of a third-order determinant. You can calculate the third-order determinant by expanding on the first row as follows

Similarly, you can calculate the third-order determinant by expanding over any row or column. It is convenient to expand the determinant along the row (or column) that contains more zeros.

Example:

Thus, the calculation of the 3rd order determinant is reduced to the calculation of 3 second order determinants. In the general case, one can calculate the determinant of a square matrix n-th order, reducing it to the calculation n determinants ( n-1)th order

Comment. Does not exist simple ways to calculate determinants of a higher order, similar to the methods for calculating determinants of the 2nd and 3rd order. Therefore, only the decomposition method can be used to calculate determinants above the third order.

Example. Calculate the fourth order determinant.

Expand the determinant by the elements of the third row

Properties of determinants:

1. The determinant will not change if its rows are replaced by columns and vice versa.

2. When permuting two adjacent rows (columns), the determinant changes sign to the opposite.

3. The determinant with two identical rows (columns) is 0.

4. Common multiplier all elements of some row (column) of the determinant can be taken out of the sign of the determinant.

5. The determinant will not change if the corresponding elements of any other column (row) multiplied by some number are added to the elements of one of its columns (rows).

Matrix determinant

Finding the determinant of a matrix is ​​a very common problem in higher mathematics and algebra. As a rule, one cannot do without the value of the matrix determinant when solving complex systems equations. Cramer's method for solving systems of equations is built on the calculation of the matrix determinant. Using the definition of a determinate, the presence and uniqueness of the solution of systems of equations are determined. Therefore, it is difficult to overestimate the importance of the ability to correctly and accurately find the determinant of a matrix in mathematics. Methods for solving determinants are theoretically quite simple, but as the size of the matrix increases, the calculations become very cumbersome and require great care and a lot of time. It is very easy to make a minor mistake or typo in such complex mathematical calculations, which will lead to an error in the final answer. Therefore, even if you find matrix determinant independently, it is important to check the result. This allows us to make our service Finding the determinant of a matrix online. Our service always gives an absolutely accurate result that does not contain any errors or typos. You can refuse independent calculations, because from the applied point of view, finding matrix determinant does not have a teaching character, but simply requires a lot of time and numerical calculations. Therefore, if in your task determination of matrix determinant are auxiliary, side calculations, use our service and find matrix determinant online!

All calculations are carried out automatically with the highest accuracy and absolutely free. We have very user-friendly interface to enter matrix elements. But the main difference between our service and similar ones is the ability to receive detailed solution. Our service at calculating matrix determinant online always uses the simplest and shortest method and describes in detail each step of the transformations and simplifications. So you get not just the value of the matrix determinant, the final result, but the whole detailed solution.

Let there be a square matrix A of size n x n .
Definition. The determinant is the algebraic sum of all possible products of elements, taken one at a time from each column and each row of the matrix A . If in each such product (term of the determinant) the factors are arranged in the order of the columns (i.e., the second indices of the elements a ij in the product are in ascending order), then with the sign (+) those products are taken for which the permutation of the first indices is even, and with the sign (-) - those for which it is odd.
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Here is the number of inversions in the permutation of indices i 1 , i 2 , …, i n .

Methods for finding determinants

1. Determinant of a matrix by expansion in rows and columns through minors.
2. Determinant by reduction to triangular form (Gauss method)

Property of determinants

1. Transposing a matrix does not change its determinant.
2. If you swap two rows or two columns of a determinant, then the determinant will change sign, but will not change in absolute value.
3. Let C = AB where A and B are square matrices. Then detC = detA ∙ detB .
4. A determinant with two identical rows or two identical columns is 0. If all elements of some row or column are equal to zero, then the determinant itself is equal to zero.
5. A determinant with two proportional rows or columns is 0.
6. The determinant of a triangular matrix is ​​equal to the product of the diagonal elements. The determinant of a diagonal matrix is ​​equal to the product of the elements on the main diagonal.
7. If all elements of a row (column) are multiplied by the same number, then the determinant will be multiplied by this number.
8. If each element of a certain row (column) of a determinant is represented as a sum of two terms, then the determinant is equal to the sum of two determinants in which all rows (columns) except the given one are the same, and in the given row (column) the first determinant contains the first ones, and in the second - the second terms.
9. Jacobi's theorem: If we add to the elements of some column of the determinant the corresponding elements of another column, multiplied by an arbitrary factor λ, then the value of the determinant will not change.

• transpose matrix;
• add to any string another string multiplied by any number.

Exercise 1. Calculate the determinant by expanding it by row or column.
Solution :xml :xls
Example 1 :xml :xls

Task 2. Calculate the determinant in two ways: a) according to the rule of "triangles"; b) string expansion.

Solution.
a) The terms included in the minus sign are constructed in the same way with respect to the secondary diagonal.

Task 3. What is the determinant of a fourth-order square matrix A if its rank is r(A)=1.
Answer: det(A) = 0.