The formula for calculating the fourth order determinant. Calculation of the determinant
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 (determinant member) 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
- Determinant of a matrix by expansion in rows and columns through minors.
- Determinant by reduction to triangular form (Gauss method)
Property of determinants
- Transposing a matrix does not change its determinant.
- If you swap two rows or two columns of a determinant, then the determinant will change sign, but will not change in absolute value.
- Let C = AB where A and B are square matrices. Then detC = detA ∙ detB .
- 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.
- A determinant with two proportional rows or columns is 0.
- 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.
- If all elements of a row (column) are multiplied by the same number, then the determinant will be multiplied by this number.
- If each element of a certain row (column) of a determinant is represented as the 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.
- 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.
| = |
b) We write the matrix in the form:
| A= |
|
Main determinant:
∆ = 2 (0 0-2 4)-(-1 (2 0-2 1))+(-2 (2 4-0 1)) = -34
Task 3. What is the determinant square matrix A is fourth order if its rank is r(A)=1.
Answer: det(A) = 0.
Determinants of the fourth and higher orders it is possible to calculate according to simplified schemes, which consist in expanding by elements of rows or columns or reducing to a triangular form. Both methods will be discussed for clarity. 4th order matrices.
Row or column decomposition method
We will consider the first example with detailed explanations all intermediate steps.
Example 1 Calculate the determinant by the expansion method.
Solution. To simplify the calculations, we expand the fourth-order determinant in terms of the elements of the first row (contains a zero element). They are formed by multiplying elements by their corresponding additions (deletions of rows and columns are formed at the intersection of the element for which they are calculated - highlighted in red) 
As a result, the calculations will be reduced to finding three third-order determinants, which we find by the rule of triangles
The found values are substituted into the output determinant
The result is easy to check with a matrix calculator YukhymCALC. To do this, select the Matrix-Matrix Determinant item in the calculator, set the matrix size to 4 * 4.
The results are the same, so the calculations are correct.
Example 2 Calculate the determinant of a matrix of the fourth order.
As in the previous task, we will carry out calculations by the decomposition method. To do this, select the elements of the first column. Simplified, the determinant can be given through the sum of four third-order determinants in the form
The calculations are not too complicated, the main thing is not to confuse with signs and triangles. We substitute the found values into the main determinant and summarize
“If you want to learn how to swim, then boldly enter the water, and if you want to learn to solve problems, then solve them.»
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(Mathematician. Made a great contribution to the popularization of mathematics. Wrote several books on how to solve problems and how to teach how to solve problems.)
Associated with each square matrix number. This number is called determinant matrices. The determinant is calculated according to special rules and is denoted by |A|, det A, ΔA.
The number of rows (columns) of a determinant is called its in order.
First order determinant matrix is equal to element a 11: |A|=a 11
Do not confuse the first order determinant with the modulus.
Second order determinant denoted by the symbol
and is equal to |A|=a 11 a 22 -a 12 a 21
3rd order determinant denoted by the symbol
To memorize this formula, schematic rules are used ( triangle rule or Sarrus)
Sarrus rule.
Triangle rule.
Let's look at an example of how these rules are used.
EXAMPLE:
Sarrus rule
We add the first two columns to the determinant.
triangle rule
This method of calculating determinants is not suitable for determinants of the 4th order and higher. Before specifying a rule that allows one to find determinants of any order, let us consider the concept of the algebraic complement of a matrix element.
Algebraic addition (And ij) element and ij matrix determinant A the number is called equal to the product(-1) i+j (to the power of the row number plus the column number of this element) by the determinant that is obtained from the given one as a result of deleting the row and column where this element stands.
EXAMPLE:
Calculate Algebraic Complement A 21 element a 21 .
SOLUTION:
By definition of algebraic complement
Calculation of the determinant of an arbitrary order. The determinant is equal to the sum of the products of the elements of any of its rows (or columns) and the corresponding algebraic additions.
, the expansion of the 4th order determinant in the first row is as follows: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: by definition 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
- For matrices of order n=2, the determinant is calculated by the formula: Δ=a 11 *a 22 -a 12 *a 21
- For matrices of order n=3, the determinant is calculated through algebraic additions or Sarrus method.
- A matrix with a dimension greater than three is decomposed into algebraic additions, for which their determinants (minors) are calculated. For instance, 4th order matrix determinant is found through expansion in rows or columns (see example).
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:- calculation of the determinant by order reduction
- calculation of the determinant by the Gaussian method (by reducing the matrix to a triangular form).
Applied use of determinants
The determinants are usually calculated for specific system, given as 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.
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 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 delete 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
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 called a number or mathematical expression, 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. The next stage of calculations is a similar multiplication of elements on the secondary diagonal and on those 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 
.
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 this element is located) - even number, an 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 us 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. The common factor of 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).
