Without matrix transformation, the determinant is easy to calculate only for 2x2 and 3x3 matrices. This is done according to the formulas:
For matrix
the determinant is:
For matrix
the determinant is:
a11*(a22*a33-a23*a32)-a12*(a21*a33-a23*a31)+a13*(a21*a32-a22*a31)
Calculations for matrices 4×4 and larger are difficult, so they need to be transformed in accordance with the properties of the determinant. It is necessary to strive to obtain a matrix in which all values except one of any column or any row are equal to zero. An example of such a matrix:
For her, the determinant is:
A12*(a21*(a33*a44-a34*a43)-a23*(a31*a44-a34*a41)+a24*(a31*a43-a33*a41))
note that
a21*(a33*a44-a34*a43)-a23*(a31*a44-a34*a41)+a24*(a31*a43-a33*a41)
this is the calculation of the determinant of the matrix obtained by subtracting the row and column, at the intersection of which there is the only non-zero number of the row/column, by which we decompose the matrix:
And we multiply the resulting value by the same number from the "zero" column / row, while the number can be multiplied by -1 (all the details are below).
If we bring the matrix to a triangular form, then its determinant is calculated as the product of the digits along the diagonal. For example, for a matrix
The determinant is:
The same should be done with matrices 5×5, 6×6 and other large dimensions.
Matrix transformations must be performed in accordance with the properties of the determinant. But before moving on to practice calculating the determinant for 4x4 matrices, let's go back to 3x3 matrices and take a closer look at how the determinant for them is calculated.
Minor
The matrix determinant is not very easy to understand, since there is a recursion in its concept: the matrix determinant consists of several elements, including the determinant of (other) matrices.
In order not to get stuck on this, let's right now (temporarily) assume that the determinant of the matrix
calculated like this:
Let's take a look at symbols and in terms like minor And algebraic addition.
The letter i denotes the ordinal number of the row, the letter j denotes the ordinal number of the column.
a ij means the matrix element (number) at the intersection of row i and column j.
Imagine a matrix that is obtained from the original one by deleting row i and column j. The determinant of the new matrix, which is obtained from the original one by deleting row i and column j, is called the minor M ij of the element a ij .
Let's illustrate what has been said. Suppose we are given a matrix
Then, to determine the minor M 11 of the element a 11, we need to create a new matrix, which is obtained from the original one by deleting the first row and the first column:
And calculate the determinant for it: 2 * 1 - (-4) * 0 = 2
To determine the minor M 22 of element a 22, we need to create a new matrix, which is obtained from the original one by removing the second row and second column:
And calculate the determinant for it: 1*1 -3*3 = -8
Algebraic addition
The algebraic complement A ij for an element a ij is the minor M ij of this element, taken with the “+” sign, if the sum of the row and column indices (i + j) at the intersection of which this element stands is even, and with the “-” sign if the sum of indices is odd.
Thus,
For the matrix from the previous example
A 11 \u003d (-1) (1 + 1) * (2 * 1 - (-4) * 0) \u003d 2
A 22 \u003d (-1) (2 + 2) * (1 * 1 -3 * 3) \u003d -8
Calculation of the determinant for matrices
The determinant of the order n corresponding to the matrix A is the number denoted det A and calculated by the formula:
In this formula, everything is already familiar to us, let's now calculate the determinant of the matrix for
Whatever the row number i=1,2,…, n or column j = 1, 2,…, n, the nth order determinant is equal to the sum of the products of the elements of this row or this column and their algebraic complements, i.e.
Those. the determinant can be calculated over any column or over any row.
To verify this, we calculate the determinant for the matrix from the last example by the second column
As you can see, the result is identical and for this matrix the determinant will always be -52, regardless of which row or column we will calculate it from.
Matrix Determinant Properties
- The rows and columns of the determinant are equal, i.e. the value of the determinant will not change if its rows and columns are swapped while maintaining their order. This operation is called the transposition of the determinant. In accordance with the stated property det A = det AT.
- When swapping two rows (or two columns), the determinant retains its absolute value, but changes sign to the opposite.
- A determinant with two identical rows (or columns) is zero.
- Multiplying all elements of some row (or some column) of the determinant by the number λ is equivalent to multiplying the determinant by the number λ.
- If all elements of any row (or any column) of the determinant are equal to zero, then the determinant itself is equal to zero.
- If the elements of two rows (or two columns) of a determinant are proportional, then the determinant is zero.
- If we add to the elements of some row (or some column) of the determinant the corresponding elements of another row (another column) multiplied by an arbitrary factor λ, then the value of the determinant will not change.
- The sum of the products of the elements of any row (any column) of the determinant and the corresponding algebraic complements of the elements of any other row (any other column) is equal to zero.
- If all i-th elements the rows of the determinant are presented as the sum of two terms a ij = b j + c j then the determinant is equal to the sum of two determinants in which all rows, except for the i-th one, are the same as in the given determinant, i-th line in one of the terms it consists of elements b j , and in the other - of elements c j . A similar property is true for determinant columns.
- The determinant of the product of two square matrices is equal to the product of their determinants: det (A * B) = det A * det B.
To calculate the determinant of any order, you can apply the method of successive reduction in the order of the determinant. To do this, use the rule of expansion of the determinant by the elements of a row or column. Another way to calculate determinants is to use elementary transformations with rows (or columns), primarily in accordance with properties 4 and 7 of the determinants, to bring the determinant to the form when under the main diagonal of the determinant (defined in the same way as for square matrices) all elements are equal to zero. Then the determinant is equal to the product of the elements located on the main diagonal.
When calculating the determinant by sequential lowering of the order in order to reduce the amount of computational work, it is advisable to use property 7 of the determinants to achieve zeroing of some of the elements of any row or any column of the determinant, which will reduce the number of calculated algebraic additions.
Reduction of a matrix to a triangular form, a matrix transformation that facilitates the calculation of the determinant
The methods shown below are not practical for 3×3 matrices, but I propose to consider the essence of the methods on simple example. Let's use the matrix for which we have already calculated the determinant - it will be easier for us to check the correctness of the calculations:
Using the 7th property of the determinant, subtract the third row from the second row, multiplied by 2:
subtract the corresponding elements of the first row of the determinant multiplied by 3 from the third row:
Since the elements of the determinant located under its main diagonal are equal to 0, then, therefore, determine is equal to the product of the elements located on the main diagonal:
1*2*(-26) = -52.
As you can see, the answer coincided with those received earlier.
Let's remember the matrix determinant formula:
The determinant is the sum of algebraic additions multiplied by the terms of one of the rows or one of the columns.
If, as a result of transformations, we make one of the rows (or a column) consist entirely of zeros except for one position, then we will not need to count all algebraic additions, since they will certainly be equal to zero. Like the previous method, this method is useful for large matrices.
Let's show an example on the same matrix:
Note that the second column of the determinant already contains one zero element. We add to the elements of the second row the elements of the first row, multiplied by -1. We get:
Let's calculate the determinant by the second column. We need to calculate only one algebraic addition, since the rest are certainly reduced to zero:
Calculation of the determinant for matrices 4×4, 5×5 and large dimensions
To avoid too large calculations for matrices of large sizes, the transformations described above should be done. Let's give a couple of examples.
Compute Decide Matrices
Solution. Using the 7th property of the determinant, we subtract the third from the second row, and from the fourth row — the corresponding elements of the first row of the determinant, multiplied by 3, 4, 5, respectively. We will abbreviate these actions as follows: (2) - (13; (3) - (1) * 4; (4) - (1) * 5. We get:
Let's take action
determinant by row or column elements
Further properties are related to the concepts of minor and algebraic complement
Definition. Minor element is called the determinant, composed of the elements remaining after deletioni-oh drains andjth column at the intersection of which this element is located. Determinant element minor n-th order has order ( n- 1). We will denote it by .
Example 1 Let 
, Then 
.
This minor is obtained from A by deleting the second row and third column.
Definition.
Algebraic addition
element is called the corresponding minor, multiplied by nat.e 
, Wherei-line number andj-columns at the intersection of which the given element is located.
VІІІ. (Decomposition of the determinant over the elements of some string). The determinant is equal to the sum of the products of the elements of some row and their corresponding algebraic additions.
.
Example 2 Let then
.
Example 3 Let's find the determinant of the matrix , expanding it by the elements of the first row.
Formally, this theorem and other properties of determinants are applicable so far only for determinants of matrices not higher than the third order, since we have not considered other determinants. The following definition will extend these properties to determinants of any order.
Definition. determinant matrices A n-th order is a number calculated using the sequential application of the decomposition theorem and other properties of determinants.
You can check that the calculation result does not depend on the order in which the above properties are applied and for which rows and columns. The determinant can be uniquely determined using this definition.
Although this definition does not contain an explicit formula for finding the determinant, it allows you to find it by reducing to determinants of matrices of lower order. Such definitions are called recurrent.
Example 4 Calculate the determinant: .
Although the decomposition theorem can be applied to any row or column of a given matrix, there will be less computation when decomposing on a column containing as many zeros as possible.
Since the matrix has no zero elements, we obtain them using property 7). Multiply the first row sequentially by the numbers (-5), (-3) and (-2) and add it to the 2nd, 3rd and 4th rows and get:
We expand the resulting determinant in the first column and get:
(take out from the 1st line (–4), from the 2nd - (–2), from the 3rd - (–1) according to property 4) 
(since the determinant contains two proportional columns).
§ 1.3. Some types of matrices and their determinants
Definition. square m an matrix that has zero elements below or above the main diagonal(=0 when i j, or =0 at i j) calledtriangular .
Their schematic structure accordingly looks like: 
or 
.
Here 0 means zero elements, and - arbitrary elements.
Theorem. The determinant of a square triangular matrix is equal to the product of its elements on the main diagonal, i.e.
.
For example:
.
Definition. A square matrix with zero elements off the main diagonal is calleddiagonal .
Its schematic view: 
A diagonal matrix that has only 1 elements on the main diagonal is called single matrix. It is denoted by:
The determinant of the identity matrix is 1, i.e. E=1.
Definition. If we choose arbitrarily k rows and k columns in the nth order determinant, then the elements at the intersection of the specified rows and columns form a square matrix of order k. The determinant is square matrix called k-th order minor .
Denoted by M k . If k=1, then the first-order minor is an element of the determinant.
The elements at the intersection of the remaining (n-k) rows and (n-k) columns make up a square matrix of order (n-k). The determinant of such a matrix is called the minor, additional to the minor M k . Denoted M n-k .
Algebraic complement of the minor M k we will call it an additional minor, taken with the “+” or “-” sign, depending on whether the sum of the numbers of all rows and columns in which the minor M k is located is even or odd.
If k=1, then the algebraic complement to the element aik calculated by the formula
A ik =(-1) i+k M ik, where M ik- minor (n-1) order.
Theorem. The product of a k-th order minor and its algebraic complement is equal to the sum of a certain number of terms of the determinant D n .
Proof
1. Consider a special case. Let the minor M k occupy the upper left corner of the determinant, that is, it is located in rows with numbers 1, 2, ..., k, then the minor M n-k will occupy rows k+1, k+2, ..., n.
Let us calculate the algebraic complement to the minor M k . A-priory,
A n-k=(-1)s M n-k, where s=(1+2+...+k) +(1+2+...+k)= 2(1+2+...+k), then
(-1) s=1 and A n-k = M n-k. Get
M k A n-k = M k M n-k. (*)
We take an arbitrary term of the minor M k
where s is the number of inversions in the substitution
and an arbitrary term of the minor M n-k
where s * is the number of inversions in the substitution
Multiplying (1) and (3), we get
The product consists of n elements located in different rows and columns of the determinant D. Therefore, this product is a member of the determinant D. The sign of the product (5) is determined by the sum of the inversions in substitutions (2) and (4), and the sign of the analogous product in the determinant D is determined by number of inversions s k in the substitution
Obviously, s k =s+s * .
Thus, returning to equality (*), we obtain that the product M k A n-k consists only of the terms of the determinant.
2. Let the minor M k located in rows with numbers i 1 , i 2 , ..., i k and in columns with numbers j 1 , j 2 , ..., j k , and i 1< i 2 < ...< i k And j1< j 2 < ...< j k .
Using the properties of determinants, with the help of transpositions, we shift the minor to the upper left corner. We obtain a determinant D ¢ in which the minor M k occupies the upper left corner, and the complementary minor M¢ n-k is the lower right corner, then, according to what was proved in paragraph 1, we get that the product M k M¢ n-k is the sum of some number of elements of the determinant D ¢ taken with its own sign. But D ¢ is obtained from D with ( i 1 -1)+(i 2 -2)+ ...+(i k -k)=(i 1 + i 2 + ...+ i k)-(1+2+...+k) string transpositions and ( j 1 -1)+(j 2 -2)+ ...+(j k -k)=(j 1 + j 2 + ...+ j k)- (1+2+...+k) column transpositions. That is, everything was done
(i 1 + i 2 + ...+ i k)-(1+2+...+k)+ (j 1 + j 2 + ...+ j k)- (1+2+...+k )= (i 1 + i 2 + ...+ i k)+ (j 1 + j 2 + ...+ j k)- 2(1+2+...+k)=s-2(1+2 +...+k). Therefore, the terms of the determinants D and D ¢ differ in sign (-1) s-2(1+2+...+k) =(-1) s , therefore, the product (-1) s M k M¢ n-k will consist of a certain number of terms of the determinant D, taken with the same signs as they have in this determinant.
Laplace's theorem. If we choose arbitrarily k rows (or k columns) 1£k£n-1 in the nth order determinant, then the sum of the products of all the kth order minors contained in the chosen rows and their algebraic complements is equal to the determinant D.
Proof
Pick random rows i 1 , i 2 , ..., i k and prove that
It was proved earlier that all elements on the left side of the equality are contained as terms in the determinant D. Let us show that each term of the determinant D falls into only one of the terms . Indeed, every t s has the form t s =. if in this product we mark the factors whose first indices i 1 , i 2 , ..., i k, and compose their product , then you can see that the resulting product belongs to the k-th order minor. Therefore, the remaining terms taken from the remaining n-k lines And n-k columns, form an element belonging to the additional minor, and taking into account the sign - to the algebraic complement, therefore, any t s falls into only one of the products , which proves the theorem.
Consequence(theorem about the expansion of the determinant in a row) . The sum of the products of the elements of some row of the determinant and the corresponding algebraic additions is equal to the determinant.
(Proof as an exercise.)
Theorem. The sum of the products of the elements of the i-th row of the determinant and the corresponding algebraic complements to the elements of the j-th row (i¹j) is equal to 0.
Comment. It is convenient to apply the corollary of Laplace's theorem to a determinant transformed using properties in such a way that in one of the rows (or in one of the columns) all elements except one are equal to 0.
Example. Compute determinant
12 -14 +35 -147 -20 -2= -160.
Matrix minors
Let a square matrix A, nth order. Minor some element a ij , matrix determinant nth order is called determinant(n - 1) -th order, obtained from the original one by deleting the row and column at the intersection of which the selected element a ij is located. Denoted M ij .
Let's look at an example matrix determinant 3 - its order:
Then according to the definition minor, minor M 12 corresponding to the element a 12 will be determinant:
At the same time, with the help minors can make it easier to calculate matrix determinant. Need to decompose matrix determinant along some line and then determinant will be equal to the sum of all elements of this row and their minors. Decomposition matrix determinant 3 - its order will look like this:
The sign before the product is (-1) n , where n = i + j.
Algebraic additions:
Algebraic addition element a ij is called its minor, taken with a "+" sign if the sum (i + j) is an even number, and with a "-" sign if this sum is an odd number. Denoted A ij . A ij \u003d (-1) i + j × M ij.
Then we can reformulate the above property. Matrix determinant is equal to the sum of the product of the elements of a certain row (row or column) matrices to their respective algebraic additions. Example:
4. Inverse matrix and its calculation.
Let A be a square matrix nth order.
Square matrix A is called non-degenerate if matrix determinant(Δ = det A) is not equal to zero (Δ = det A ≠ 0). Otherwise (Δ = 0) matrix It's called degenerate.
Matrix, allied to matrix Ah, it's called matrix
Where A ij - algebraic addition element a ij given matrices(it is defined in the same way as algebraic addition element matrix determinant).
Matrix A -1 is called inverse matrix A, if the condition is met: A × A -1 \u003d A -1 × A \u003d E, where E is a single matrix the same order as matrix A. Matrix A -1 has the same dimensions as matrix A.
inverse matrix
If there are square matrices X and A satisfying the condition: X × A \u003d A × X \u003d E, where E is the unit matrix the same order, then matrix X is called inverse matrix to the matrix A and is denoted by A -1 . Any non-degenerate matrix It has inverse matrix and moreover, only one, i.e., in order to square matrix A had inverse matrix, it is necessary and sufficient that it determinant was different from zero.
For getting inverse matrix use the formula:
Where M ji is optional minor element a ji matrices A.
5. Matrix rank. Calculation of the rank using elementary transformations.
Consider a rectangular matrix mxn. Let's single out some k rows and k columns in this matrix, 1 £ k £ min (m, n) . From the elements at the intersection of the selected rows and columns, we will compose the determinant of the kth order. All such determinants are called matrix minors. For example, for a matrix, you can compose second-order minors 
and first order minors 1, 0, -1, 2, 4, 3.
Definition. The rank of a matrix is the highest order of the non-zero minor of this matrix. Denote the rank of the matrix r (A).
In the above example, the rank of the matrix is two, since, for example, the minor
The rank of a matrix is conveniently calculated by the method of elementary transformations. The elementary transformations include the following:
1) permutations of rows (columns);
2) multiplying a row (column) by a non-zero number;
3) adding to the elements of a row (column) the corresponding elements of another row (column), previously multiplied by a certain number.
These transformations do not change the rank of the matrix, since it is known that 1) when the rows are permuted, the determinant changes sign and, if it was not equal to zero, then it will not; 2) when multiplying the row of the determinant by a number that is not equal to zero, the determinant is multiplied by this number; 3) the third elementary transformation does not change the determinant at all. Thus, by performing elementary transformations on the matrix, one can obtain a matrix for which it is easy to calculate the rank of it and, consequently, of the original matrix.
Definition. A matrix obtained from a matrix using elementary transformations is called equivalent and is denoted A IN.
Theorem. The rank of a matrix does not change under elementary matrix transformations.
With the help of elementary transformations, one can bring the matrix to the so-called step form, when the calculation of its rank is not difficult.
Matrix is called stepped if it has the form:
Obviously, the rank of the step matrix is equal to the number of non-zero rows , because there is a minor of the th order, not equal to zero:
.
Example. Determine the rank of a matrix using elementary transformations.
The rank of a matrix is equal to the number of non-zero rows, i.e. .
Algebraic addition- the concept of matrix algebra; in relation to the element aij of the square matrix A is formed by multiplying the minor of the element aij by (1)i+j; denoted by Aij: Aij=(1)i+jMij, where Mij is the minor of the element aij of the matrix A=, i.e. determinant ... ... Economic and Mathematical Dictionary
algebraic addition- The concept of matrix algebra; in relation to the element aij of the square matrix A is formed by multiplying the minor of the element aij by (1)i+j; denoted by Aij: Aij=(1)i+jMij, where Mij is the minor of the element aij of the matrix A=, i.e. matrix determinant, ... ... Technical Translator's Handbook
See Art. Determinant ... Great Soviet Encyclopedia
For a minor M, a number equal to where M is a minor of order k, located in rows with numbers and columns with numbers of some square matrix A of order n; determinant of a matrix of order n k, obtained from matrix A by deleting rows and columns of minor M; ... ... Mathematical Encyclopedia
Wiktionary has an entry for "Addition" Addition may refer to ... Wikipedia
The operation, to the edge, associates a subset M of a given set X with another subset so that if Mi N are known, then the set X can be restored in one way or another. Depending on what structure the set X is endowed with, ... ... Mathematical Encyclopedia
Or a determinant, in mathematics, a record of numbers in the form of a square table, in accordance with which another number (the value of the determinant) is put. Very often, the concept of a determinant means both the meaning of the determinant and the form of its notation. ... ... Collier Encyclopedia
For a theorem from the theory of probability, see the article Local theorem of Moivre Laplace. Laplace's theorem is one of the theorems of linear algebra. Named after the French mathematician Pierre Simon Laplace (1749 1827), who is credited with formulating ... ... Wikipedia
- (Laplacian matrix) one of the representations of a graph using a matrix. The Kirchhoff matrix is used to count the spanning trees of a given graph (matrix tree theorem) and is also used in spectral graph theory. Contents 1 ... ... Wikipedia
An equation is a mathematical relation expressing the equality of two algebraic expressions. If the equality is true for any admissible values of the unknowns included in it, then it is called an identity; for example, the ratio of the species ... ... Collier Encyclopedia
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- Discrete Mathematics, A. V. Chashkin. 352 pages. The textbook consists of 17 chapters on the main sections of discrete mathematics: combinatorial analysis, graph theory, boolean functions, computational complexity and coding theory. Contains…
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