Elements of the theory of determinants and matrices. Abstract: Theory of Matrices and Determinants

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Elements of determinant theory

A determinant is a number written in the form of a square table of numbers, calculated according to certain rules.

For example, each of the tables (1.1) consists of an equal number of rows and columns and represents a number, the calculation rules for which will be discussed below.

The number of rows and columns determines the order of the determinant. Thus, determinant 1.1a) is of the third order, determinant 1.1b) is of the second order, 1.1c) is of the first order. As you can see, the first-order determinant is the number itself.

Straight vertical brackets at the edges of the table are the sign and symbol of the determinant. Is the determinant indicated by a capital letter of the Greek alphabet? (delta).

In general form, the nth order determinant is written as follows:

Each element A ij the determinant has two indices: the first index i indicates the line number, second j- number of the column at the intersection of which the element is located. So for determinant 1.1a) elements A 11 , A 22 , A 23 , A 32 are respectively equal to 2, 5, 4, 3.

The 2nd order determinant is calculated using the formula

The 2nd order determinant is equal to the product of the elements on the main diagonal minus the product of the elements on the secondary diagonal.

To calculate the 3rd order determinant, the “triangle method” and the Sarrus method are used. But usually in practice, to calculate the 3rd order determinant, the so-called method of effective order reduction is used, which will be discussed below.

Triangle method

When calculating the determinant using this method, it is convenient to use its graphical representation. In Fig. 1.1 and 1.2, the elements of the 3rd order determinant are schematically represented by dots.

Rice. 1.1 Fig. 1.2

When calculating the determinant, the product of elements connected by straight lines follows the diagram in Fig. 1.1, take with a plus sign, and the product of elements connected according to the diagram in Fig. 1.2, take with a minus sign. As a result of these actions, the formula used for the calculation takes the form:

Calculate the 3rd order determinant.

Sarrus method

To implement it, you need to assign the first two columns to the right of the determinant, compose the products of the elements located on the main diagonal and on lines parallel to it, and take them with a plus sign. Then compose the products of elements located on the side diagonal and parallel to it with a minus sign.

Scheme for calculating the determinant using the Sarrus method.

Calculate the determinant given in Example 1.2 using the Sarrus method.

Minor and algebraic complement of the determinant element

Minor M ij element A ij is called the determinant ( n-1) -th order obtained from the determinant n-th order by crossing out i-th line and j th column (i.e. by crossing out the row and column at the intersection of which the element is located A ij).

Find the minor of elements A 23 And A 34 determinant of the 4th order.

Element A 23 is in the 2nd row and 3rd column. In this example A 23 =4. Crossing out the 2nd row and 3rd column at the intersection of this element (shown for methodological purposes by vertical and horizontal dotted lines), we obtain the minor M 23 of this element. This will already be a 3rd order determinant.

When calculating minors, the operation of crossing out a row and column is performed mentally. Having done this, we get

Algebraic complement A ij element A ij determinant n The th order is the minor of this element, taken with the sign (-1) i + j, Where i+ j- the sum of the row and column numbers to which the element belongs A ij. Those. a-priory A ij=(-1) i + jM ij

It is clear that if the amount i+ j- the number is even, then A ij=M ij, If i+ j- the number is odd, then A ij= - M ij.

For the determinant, find the algebraic complements of the elements A 23 And A 31 .

For element A 23 i=2, j=3 and i+ j=5 is an odd number, hence

For element A 31 i=3, j=1 and i+ j=4 is an even number, which means

Properties of determinants

1. If any two parallel rows (two rows or two columns) are swapped in the determinant, the sign of the determinant changes to the opposite

Swap 2 parallel columns (1st and 2nd).

Swap 2 parallel lines (1st and 3rd).

2. The common factor of the elements of any row (row or column) can be taken out of the determinant sign.

Properties of a determinant being equal to zero

3. If all elements of a certain series in a determinant are equal to zero, such a determinant is equal to zero.

4. If in a determinant the elements of any series are proportional to the elements of a parallel series, the determinant is equal to zero.

Properties of invariance (immutability) of the determinant.

5. If the rows and columns in the determinant are swapped, the determinant will not change.

6. The determinant will not change if elements of any parallel series are added to the elements of any series, first multiplying by a certain number.

Property 6 is widely used in calculating determinants using the so-called effective order reduction method. When applying this method, it is necessary to bring all elements except one to zero in one row (one row or column). A non-zero element of the determinant will be equal to zero if it is added to a number of equal magnitude but opposite sign.

Let's show with an example how this is done.

Using properties 2 and 6, reduce the determinant to a determinant that has two zeros in any row.

Using property 2, we simplify the determinant by removing 2 from the 1st row, 4 from the 2nd row and 2 from the 3rd row as common factors.

Because element A 22 is equal to zero, then to solve the problem it is enough to reduce any element in the 2nd row or 2nd column to zero. There are several ways to do this.

For example, let's take the element A 21 =2 to zero. To do this, based on property 6, multiply the entire third column by (-2) and add it to the first. Having performed this operation, we get

It is possible to null an element A 12 =2, then we will get two elements equal to zero in the second column. To do this, you need to multiply the 3rd line by (-2) and add the resulting values to the first line

Calculation of determinant of any order

The rule for calculating the determinant of any order is based on Laplace's theorem.

Laplace's theorem

The determinant is equal to the sum of pairwise products of the elements of any row (row or column) by their algebraic complements.

According to this theorem, the determinant can be calculated by decomposing it either over the elements of any row or any column.

In general, the nth order determinant can be expanded and calculated in the following ways:

Calculate the determinant using Laplace's theorem by decomposing it into the elements of the 3rd row and the elements of the 1st column.

We calculate the determinant by expanding it along the 3rd line

Let's calculate the determinant by expanding it over the first column

Effective order reduction method

The complexity of calculating the determinant using Laplace's theorem will be significantly less if there is only one term in its expansion either in a row or in a column. Such an expansion will be obtained if in the row (or column) along which the determinant is expanded, all elements except one are equal to zero. The method of “zeroing” the elements of the determinant was discussed earlier.

Calculate the determinant using the effective order reduction method.

Because determinant of the 3rd order, then we “zero” any 2 elements of the determinant. It is convenient for this purpose to take the 2nd column, the element of which A 22 = - 1. In order for the element A 21 was equal to zero, the 1st column should be added to the 2nd. In order for the element A 23 was equal to zero, you need to multiply the 2nd column by 2 and add it to the 3rd. After performing these operations, the given determinant is converted to the determinant

Now we expand this determinant along the 2nd line

Calculation of the determinantcutting it into a triangular shape

A determinant for which all elements above or below the main diagonal are equal to zero is called a triangular determinant. In this case, the determinant is equal to the product of its elements of the main diagonal.

Reducing the determinant to triangular form is always possible based on its properties.

A determinant is given. Reduce it to triangular form and calculate.

Let's "zero out", for example, all elements located above the main diagonal. To do this, you need to perform three operations: 1st operation - add the first line with the last, we get A 13 = 0. 2nd operation - multiplying the last line by (-2) and adding with the 2nd, we get A 23 = 0. The sequential execution of these operations is shown below.

To reset an element A 12 add the 1st and 2nd lines

Elements of matrix theory

A matrix is a table of numbers or any other elements containing m lines and n columns.

General view of the matrix

The matrix, like the determinant, has elements equipped with a double index. The meaning of indices is the same as for determinants.

If the determinant is equal to a number, then the matrix is not equated to any other simpler object.

The parentheses on the sides of the matrix are its sign or symbol (but not the straight brackets that denote the determinant). For brevity, the matrix is denoted in capital letters A, B, C etc.

A matrix has a size that is determined by its number of rows and columns, which is written as - A m n.

For example, a numeric matrix of size 23 has the form, size 31 has the form, size 14 has the form, etc.

A matrix in which the number of rows is equal to the number of columns is called square. In this case, as for determinants, we talk about the order of the matrix.

For example, a 3rd order numerical matrix has the form

Types of matrices

A matrix consisting of one row is called a row matrix

A matrix consisting of one column is called a column matrix

The matrix is called square n-th order if the number of its rows is equal to the number of columns and is equal to n.

For example, a square matrix of 3rd order.

A diagonal matrix is a square matrix in which all elements are zero except those on the main diagonal. The main diagonal is the diagonal that runs from the top left corner to the bottom right corner.

For example, a third-order diagonal matrix.

A diagonal matrix, all elements of which are equal to one, is called identity and is denoted by the letter E or number 1

A null matrix is a matrix in which all elements are equal to zero.

An upper triangular matrix is a matrix in which all elements located below the main diagonal are equal to zero.

A lower triangular matrix is a matrix in which all elements located above the main diagonal are equal to zero.

For example

Upper triangular matrix

Lower triangular matrix

If in the matrix A swap rows with columns, we get a transposed matrix, which is denoted by the symbol A*.

For example, given a matrix,

matrix transposed with respect to it A*

Square matrix A has a determinant, which is denoted by det A(det is a shortened French word for "determiner").

For example, for the matrix A

we write down its determinant

All operations with the determinant of a matrix are the same as discussed earlier.

A matrix whose determinant is equal to zero is called special, or degenerate, or singular. A matrix for which its determinant is not equal to zero is called non-singular or non-singular.

Union or annexed matrix.

If for a given square matrix A determine the algebraic complements of all its elements and then transpose them, then the matrix thus obtained will be called allied or adjoint to the matrix A and is indicated by the symbol A

For a matrix find A.

Compiling the determinant of the matrix A

We determine the algebraic complements of all elements of the determinant using the formula

Transposing the resulting algebraic complements, we obtain the allied or adjoint matrix A in relation to a given matrix A.

Actions on matrices

Matrix equality

Two matrices A And IN are considered equal if:

a) they both have the same size;

b) the corresponding elements of these matrices are equal to each other. Corresponding elements are elements with the same indices.

Addition and subtraction of matrices

You can only add and subtract matrices of the same dimension. The sum (difference) of two matrices A And IN there will be a third matrix WITH, whose elements WITH ij equal to the sum (difference) of the corresponding matrix elements A And IN. According to the definition, matrix elements WITH are according to the rule.

For example, if

The concept of a sum (difference) of matrices extends to any finite number of matrices. In this case, the sum of matrices obeys the following laws:

a) commutative A + B = B + A;

b) associative WITH + (A + B) = (B + C)+ A.

Multiplying a matrix by a number.

To multiply a matrix by a number, you need to multiply each element of the matrix by that number.

Consequence. The common factor of all matrix elements can be taken out of the matrix sign.

For example, .

As you can see, the actions of adding, subtracting matrices, and multiplying a matrix by a number are similar to actions on numbers. Matrix multiplication is a specific operation.

Product of two matrices.

Not all matrices can be multiplied. Product of two matrices A And IN in the order listed A IN only possible when the number of columns of the first factor A equal to the number of rows of the second factor IN.

For example, .

Matrix size A 33, matrix size IN 23. Work A IN impossible, work IN A Maybe.

The product of two matrices A and B is the third matrix C, the element C ij of which is equal to the sum of pairwise products of the elements of the i-th row of the first factor and the j-th column of the second factor.

It was shown that in this case the product of matrices is possible IN A

From the rule of existence of the product of two matrices it follows that the product of two matrices in the general case does not obey the commutative law, i.e. A IN? IN A. If in a particular case it turns out that A B = B A, then such matrices are called permutable or commutative.

In matrix algebra, the product of two matrices can be a zero matrix even when none of the factor matrices is zero, contrary to ordinary algebra.

For example, let's find the product of matrices A IN, If

You can multiply multiple matrices. If you can multiply matrices A, IN and the product of these matrices can be multiplied by the matrix WITH, then it is possible to compose the product ( A IN) WITH And A(IN WITH). In this case, the combinational law regarding multiplication takes place ( A IN) WITH = A(IN WITH).

inverse matrix

If two matrices A And IN the same size, and their product A IN is the identity matrix E, then matrix B is called the inverse of A and is denoted A -1 , i.e. A A -1 = E.

inverse matrix A -1 equal to the ratio of the union matrix A to the determinant of the matrix A

From this it is clear that in order for the inverse matrix to exist A -1 it is necessary and sufficient that the matrix det A? 0, i.e., so that the matrix A was non-degenerate.

For a matrix find A -1 .

Determining the value of the determinant of the matrix A

Because det A? 0, the inverse matrix exists. In example 2.1. for a given determinant the allied matrix was found

A-priory

Matrix rank

For solving and studying a number of mathematical and applied problems, the concept of matrix rank is important.

Consider the matrix A size m n

Select randomly in the matrix Ak lines and k columns. Elements located at the intersection of selected rows and columns form a square matrix k-of that order. The determinant of this matrix is called the minor k-order of matrix A. Select k lines and k columns can be used in different ways, resulting in different minors k-of that order. The 1st order minors are the elements themselves. Obviously, the largest possible order of minors is equal to the smallest of the numbers m And n. Among the formed minors of various orders there will be those that are equal to zero and not equal to zero.

Highest order of nonzero matrix minors A is called the rank of the matrix.

Matrix rank A denoted by rank A or r( A).

If the matrix rank A equals r, then this means that the matrix has a non-zero minor of order r, but every minor is of greater order than r equal to zero.

From the definition of matrix rank it follows that:

a) matrix rank A m n does not exceed the smaller of its sizes, i.e. r(A) ? min(m, n);

b) r(A) = 0 if and only if all elements of the matrix are equal to zero, i.e. A = 0;

c) for a square matrix n-th order r(A) = n, if the matrix is non-singular.

Let's look at an example of determining the rank of a matrix using the method of bordering minors. Its essence lies in sequentially enumerating the minors of the matrix and finding the highest order non-zero minor.

Calculate the rank of the matrix.

For matrix A 3 4 r(A) ? min (3,4) = 3. Let's check whether the rank of the matrix is equal to 3; to do this, we calculate all the third-order minors (there are only 4 of them, they are obtained by deleting one of the columns of the matrix).

Since all third order minors are zero, r(A) ? 2. Since there is a zero minor of the second order, for example

That r(A) = 2.

Any non-zero minor of a matrix whose order is equal to its rank is called a basis minor of this matrix.

A matrix can have more than one basis minor, but several. However, the orders of all basis minors are the same and equal to the rank of the matrix.

The rows and columns that form a basis minor are called basis.

Every row (column) of a matrix is a linear combination of the basis rows (columns).

1. Matrices.

1.1 The concept of a matrix.

Matrix is a rectangular table of numbers containing a certain quantity m lines and a certain number n columns. Numbers m And n are called orders matrices. If m = n , the matrix is called square, and the number m = n - her in order .

1.2 Basic operations on matrices.

The basic arithmetic operations on matrices are multiplying a matrix by a number, adding and multiplying matrices.

Let's move on to defining the basic operations on matrices.

Matrix addition : The sum of two matrices, for example: A And B , having the same number of rows and columns, in other words, the same orders m And n called matrix C = ( WITH ij )( i = 1, 2, …m; j = 1, 2, …n) the same orders m And n , elements Cij which are equal.

Cij = Aij + Bij (i = 1, 2, …, m; j = 1, 2, …, n) ( 1.2 )

To denote the sum of two matrices, the notation is used C = A + B. The operation of summing matrices is called their addition

So by definition we have:

+ =

From the definition of the sum of matrices, or more precisely from the formula ( 1.2 ) it immediately follows that the operation of adding matrices has the same properties as the operation of adding real numbers, namely:

commutative property: A + B = B + A

combining property: (A + B) + C = A + (B + C)

These properties make it possible not to worry about the order of the matrix terms when adding two or more matrices.

Multiplying a matrix by a number :

Matrix product to a real number called a matrix C = (Cij) (i = 1, 2, … , m; j = 1, 2, …, n) , whose elements are equal

Cij = Aij (i = 1, 2, …, m; j = 1, 2, …, n). ( 1.3 )

To denote the product of a matrix and a number, the notation is used C= A or C=A . The operation of composing the product of a matrix by a number is called multiplying the matrix by this number.

Directly from formula ( 1.3 ) it is clear that multiplying a matrix by a number has the following properties:

distributive property regarding the sum of matrices:

( A + B) = A+ B

associative property regarding a numerical factor:

( ) A= ( A)

distributive property regarding the sum of numbers:

( + ) A= A + A .

Comment : Difference of two matrices A And B of identical orders it is natural to call such a matrix C of the same orders, which in sum with the matrix B gives the matrix A . To denote the difference between two matrices, a natural notation is used: C = A – B.

Matrix multiplication :

Matrix product A = (Aij) (i = 1, 2, …, m; j = 1, 2, …, n) , having orders respectively equal m And n , per matrix B = (Bij) (i = 1, 2, …, n;

j = 1, 2, …, p) , having orders respectively equal n And p , is called a matrix C= (WITH ij) (i = 1, 2, … , m; j = 1, 2, … , p) , having orders correspondingly equal m And p , and elements Cij , defined by the formula

Cij = (i = 1, 2, …, m; j = 1, 2, …, p) ( 1.4 )

To denote the product of a matrix A to the matrix B use recording

C=AB . The operation of composing a matrix product A to the matrix B called multiplication these matrices. From the above definition it follows that matrix A cannot be multiplied by any matrix B : it is necessary that the number of matrix columns A was equals number of matrix rows B . In order for both works AB And B.A. were not only defined, but also had the same order, it is necessary and sufficient that both matrices A And B were square matrices of the same order.

Formula ( 1.4 ) is a rule for composing matrix elements C ,

which is the product of the matrix A to the matrix B . This rule can be formulated verbally: Element Cij , standing at the intersection i th line and j- th matrix column C=AB , is equal the sum of pairwise products of the corresponding elements i th line matrices A And j- th matrix column B . As an example of the application of this rule, we present the formula for multiplying square matrices of the second order

From the formula ( 1.4 ) the following properties of the matrix product follow: A to the matrix B :

associative property: ( AB) C = A(BC);

distributive property with respect to the sum of matrices:

(A + B) C = AC + BC or A (B + C) = AB + AC.

It makes sense to raise the question of the permutation property of a product of matrices only for square matrices of the same order. Elementary examples show that products of two square matrices of the same order does not, generally speaking, have the commutation property. In fact, if we put

A= , B = , That AB = , A BA =

The same matrices for which the product has the commutation property are usually called commuting.

Among square matrices, we highlight the class of so-called diagonal matrices, each of which has elements located outside the main diagonal equal to zero. Among all diagonal matrices with coinciding elements on the main diagonal, two matrices play a particularly important role. The first of these matrices is obtained when all elements of the main diagonal are equal to one, and is called the identity matrix n- E . The second matrix is obtained with all elements equal to zero and is called the zero matrix n- order and is denoted by the symbol O . Let us assume that there is an arbitrary matrix A , Then

AE=EA=A , AO=OA=O .

The first of the formulas characterizes the special role of the identity matrix E , similar to the role played by the number 1 when multiplying real numbers. As for the special role of the zero matrix ABOUT , then it is revealed not only by the second of the formulas, but also by an elementary verifiable equality: A+O=O+A=A . The concept of a zero matrix can be introduced not for square matrices.

2. Determinants.

2.1 The concept of a determinant.

First of all, you need to remember that determinants exist only for matrices of square type, because there are no determinants for matrices of other types. In the theory of systems of linear equations and in some other issues, it is convenient to use the concept determinant , or determinant .

2.2 Calculation of determinants.

Consider any four numbers written in the form of a matrix two in lines and each two columns , Determinant or determinant , made up of the numbers in this table, is the number ad-bc , denoted as follows: . Such a determinant is called second order determinant , since a table of two rows and two columns was taken to compile it. The numbers that make up the determinant are called its elements ; at the same time they say that the elements a And d make up main diagonal determinant, and the elements b And c his side diagonal . It can be seen that the determinant is equal to the difference of the products of pairs of elements located on its main and secondary diagonals. The determinant of the third and any other order is approximately the same, namely: Let's say we have a square matrix . The determinant of the following matrix is the following expression: a11a22a33 + a12a23a31 + a13a21a32 – a11a23a32 – a12a21a33 – a13a22a31. . As you can see, it is calculated quite easily if you remember a certain sequence. With a positive sign are the main diagonal and the triangles formed from the elements, which have a side parallel to the main diagonal, in this case these are triangles a12a23a31 , a13a21a32 .

The side diagonal and the triangles parallel to it have a negative sign, i.e. a11a23a32, a12a21a33 . In this way determinants of any order can be found. But there are cases when this method becomes quite complicated, for example, when there are a lot of elements in the matrix, and in order to calculate the determinant you need to spend a lot of time and attention.

There is an easier way to calculate the determinant n- oh order, where n 2 . Let's agree to call any element a minor Aij matrices n- first-order determinant corresponding to the matrix that is obtained from the matrix as a result of deleting i th line and j- th column (that row and that column at the intersection of which there is an element Aij ). Element minor Aij we will denote by the symbol . In this notation, the upper index denotes the row number, the lower index the column number, and the bar over M means that the specified row and column are crossed out. Determinant of order n , corresponding to the matrix, we call the number equal to and denoted by the symbol .

Theorem 1.1 Whatever the line number i ( i =1, 2…, n) , for the determinant n- the first order of magnitude formula is valid

= det A =

called i- th line . We emphasize that in this formula the exponent to which the number is raised (-1) is equal to the sum of the row and column numbers at the intersection of which the element is located Aij .

Theorem 1.2 Whatever the column number j ( j =1, 2…, n) , for the determinant n the th order formula is valid

= det A =

called expansion of this determinant in j- th column .

2.3 Basic properties of determinants.

Determinants also have properties that make the task of calculating them easier. So, below we establish a number of properties that an arbitrary determinant has n -th order.

1 . Row-column equality property . Transposition of any matrix or determinant is an operation that results in the rows and columns being swapped while preserving their order. As a result of matrix transposition A the resulting matrix is called a matrix, called transposed with respect to the matrix A and is indicated by the symbol A .

The first property of the determinant is formulated as follows: during transposition, the value of the determinant is preserved, i.e. = .

2 . Antisymmetry property when rearranging two rows (or two columns) . When two rows (or two columns) are swapped, the determinant retains its absolute value, but changes sign to the opposite. For a second-order determinant, this property can be verified in an elementary way (from the formula for calculating the second-order determinant it immediately follows that the determinants differ only in sign).

3 . Linear property of the determinant. We will say that some string ( a) is a linear combination of the other two strings ( b And c ) with coefficients And . The linear property can be formulated as follows: if in the determinant n -th order some i The -th row is a linear combination of two rows with coefficients And , That = + , Where

– determinant that has i The -th row is equal to one of the two rows of the linear combination, and all other rows are the same as , A - a determinant that has i- i string is equal to the second of the two strings, and all other strings are the same as .

These three properties are the main properties of the determinant, revealing its nature. The following five properties are logical consequences three main properties.

Corollary 1. A determinant with two identical rows (or columns) is equal to zero.

Corollary 2. Multiplying all elements of some row (or some column) of a determinant by a number a is equivalent to multiplying the determinant by this number a . In other words, the common factor of all elements of a certain row (or some column) of a determinant can be taken out of the sign of this determinant.

Corollary 3. If all elements of a certain row (or some column) are equal to zero, then the determinant itself is equal to zero.

Corollary 4. If the elements of two rows (or two columns) of a determinant are proportional, then the determinant is equal to zero.

Corollary 5. If to the elements of a certain row (or some column) of the determinant we add the corresponding elements of another row (another column), multiplication by an arbitrary factor , then the value of the determinant does not change. Corollary 5, like the linear property, allows for a more general formulation, which I will give for strings: if to the elements of a certain row of a determinant we add the corresponding elements of a string that is a linear combination of several other rows of this determinant (with any coefficients), then the value of the determinant will not change . Corollary 5 is widely used in the concrete calculation of determinants.

3. Systems of linear equations.

3.1 Basic definitions.

…….

3.2 Condition for compatibility of systems of linear equations.

…….

3.3 Solving systems of linear equations using the Cramer method.

It is known that using matrices we can solve various systems of equations, and these systems can be of any size and have any number of variables. With a few derivations and formulas, solving huge systems of equations becomes quite fast and easier.

In particular, I will describe the Cramer and Gauss methods. The easiest way is the Cramer method (for me), or as it is also called, the Cramer formula. So, let's say that we have some system of equations . The main determinant, as you have already noticed, is a matrix made up of the coefficients of the variables. They also appear in column order, i.e. the first column contains the coefficients that are found at x , in the second column at y , and so on. This is very important, because in the following steps we will replace each column of coefficients for a variable with a column of equation answers. So, as I said, we replace the column at the first variable with the answer column, then at the second, of course it all depends on how many variables we need to find.

1 = , 2 = , 3 = .

Then you need to find determinants determinant of the system .

3.4 Solving systems of linear equations using the Gauss method.

…….

4. Inverse matrix.

4.1 The concept of an inverse matrix.

4.2 Calculation of the inverse matrix.

Bibliography.

V. A. Ilyin, E. G. Poznyak “Linear Algebra”

2. G. D. Kim, E. V. Shikin “Elementary transformations in linear algebra”

Topic 1. Matrices and matrix determinants

What we learn:

Basic concepts of linear algebra: matrix, determinant.

What we will learn:

Perform operations on matrices;

Calculate with second and third order determinants.

Topic 1.1. The concept of a matrix. Actions on matrices

∆ Matrix is a rectangular table consisting of rows and columns filled with some mathematical objects.

Matrices are denoted in capital Latin letters, the table itself is enclosed in parentheses (less often in square or other shapes).

Elements A ij called matrix elements . First index i– line number, secondj– column number. Most often the elements are numbers.

Entry "matrix" A has the size m× n» means that we are talking about a matrix consisting ofm lines and n columns.

If m = 1, a n > 1, then the matrix ismatrix - row . If m > 1, a n = 1, then the matrix ismatrix - column .

A matrix in which the number of rows coincides with the number of columns (m= n), called square .

Elements a 11 , a 22 ,…, a nn square matrixA (size n× n) form main diagonal , elements a 1 n , a 2 n -1 ,…, a n 1 - side diagonal .

In the matrix
elements 5; 7 form the main diagonal, elements –5; 8 – side diagonal.

Matrices A And B are called equal (A= B), if they have the same size and their elements in the same positions coincide, i.e.A ij = b ij .

Identity matrix is a square matrix in which the elements of the main diagonal are equal to one, and the remaining elements are equal to zero. The identity matrix is usually denoted E.

Matrix transposed to matrix A of sizem× n, is called matrix A T size n× m, obtained from matrix A, if its rows are written into columns, and its columns into rows.

Arithmetic operations on matrices.

To find sum of matrices A And B of the same dimension, it is necessary to add elements with the same indices (standing in the same places):

Matrix addition is commutative, that is, A + B = B + A.

To find matrix difference A And B of the same dimension, it is necessary to find the difference of elements with the same indices:

To multiply matrix Aper number k, It is necessary to multiply each element of the matrix by this number:

Work matrices AB can only be defined for matricesA size m× n And B size n× p, i.e. number of matrix columnsA must be equal to the number of matrix rowsIN. Wherein A· B= C, matrix C has the size m× p, and its element c ij is found as a scalar productith matrix rows A on jth matrix columnB: ( i=1,2,…, m; j=1,2,…, p).

!! Actually every line is needed matrices A (standing on the left) multiply scalarly by each matrix column B (standing on the right).

The product of matrices is not commutative, that isА·В ≠ В·А . ▲

☺ It is necessary to analyze examples to consolidate theoretical material.

Example 1. Determining the size of matrices.

Example 2. Definition of matrix elements.

In the matrix element A 11 = 2, A 12 = 5, A 13 = 3.

In the matrix element A 21 = 2, A 13 = 0.

Example 3: Performing matrix transposition.

Example 4. Performing operations on matrices.

Find 2 A- B, If , .

Solution. .

Example 5. Find the product of matrices And .

Solution. Matrix sizeA – 3 × 2 , matrices IN – 2 × 2 . Therefore the productA·B you can find it. We get:

Work VA can't be found.

Example 6. Find A 3 if A =
.

Solution. A 2 = ·=
=
,

A 3 = ·=
=
.

Example 6. Find 2 A 2 + 3 A + 5 E at
,
.

Solution. ,

,
,

,
.

Tasks to complete

1. Fill out the table.

Matrix	Size	Matrix type	Matrix elements
Matrix	Size	Matrix type		a 12	a 23	a 32	a 33

2. Perform operations on matrices
And
:

3. Perform matrix multiplication:

4. Transpose matrices:

? 1. What is a matrix?

2. How to distinguish a matrix from other elements of linear algebra?

3. How to determine the matrix size? Why is this necessary?

4. What does the entry mean? A ij ?

5. Give an explanation of the following concepts: main diagonal, secondary diagonal of the matrix.

6. What operations can be performed on matrices?

7. Explain the essence of the matrix multiplication operation?

8. Can any matrices be multiplied? Why?

Topic 1.2. Second and third order determinants : m methods for their calculation

∆ If A is a square matrix n-th order, then we can associate with it a number called determinant nth order and denoted by |A|. That is, the determinant is written as a matrix, but instead of parentheses it is enclosed in straight brackets.

!! Sometimes determiners are called determinants in the English manner, that is = det A.

1st order determinant (determinant of matrix A of size1 × 1 ) is the element itself that matrix A contains, that is.

2nd order determinant (matrix determinant A size 2 × 2 ) is a number that can be found using the rule:

(the product of the elements on the main diagonal of the matrix minus the product of the elements on the secondary diagonal).

3rd order determinant (matrix determinant A size 3 × 3 ) is a number that can be found using the “triangles” rule:

To calculate 3rd order determinants, you can use a simpler rule - the rule of directions (parallel lines).

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

!! To calculate determinants, you can use their properties, which are valid for determinants of any order.

Properties of determinants:

1° . The determinant of matrix A does not change during transposition, i.e. |A| = |A T |. This property characterizes the equality of rows and columns.

2° . When rearranging two rows (two columns), the determinant retains its previous value, but the sign is reversed.

4° . If any row or column contains a common factor, then it can be taken out of the determinant sign.

Corollary 4.1. If all elements of any series of a determinant are equal to zero, then the determinant is equal to zero.

Corollary 4.2. If the elements of any series of a determinant are proportional to the corresponding elements of a series parallel to it, then the determinant is equal to zero.▲

☺ It is necessary to analyze the rules for calculating determinants.

Example 1: Calculationsecond order determinants,
.

Solution.

Secondary school No. 45.

Moscow city.

Student of 10th grade “B” Gorokhov Evgeniy

Coursework (draft).

Introduction to the theory of matrices and determinants .

1. Matrices........................................................ ........................................................ ........................................................ ......

1.1 Concept of matrix.................................................... ........................................................ ...................................

1.2 Basic operations on matrices.................................................... ........................................................ .

2. Determinants........................................................ ........................................................ ...........................................

2.1 The concept of a determinant.................................................... ........................................................ ........................

2.2 Calculation of determinants.................................................... ........................................................ ...............

2.3 Basic properties of determinants.................................................... ........................................................

3. Systems of linear equations.................................................... ........................................................ .

3.1 Basic definitions................................................... ........................................................ ........................

3.2 Consistency condition for systems of linear equations.................................................... ...............

3.3 Solving systems of linear equations using Cramer's method.................................................... ..........

3.4 Solving systems of linear equations using the Gaussian method.................................................... .............

4. Inverse matrix.................................................... ........................................................ ...................................

4.1 Concept of inverse matrix.................................................... ........................................................ ................

4.2 Calculation of the inverse matrix.................................................... ........................................................ ........

Bibliography................................................ ........................................................ ................................

Matrix is a rectangular table of numbers containing a certain quantity m lines and a certain number n columns. Numbers m And n are called orders matrices. If m = n , the matrix is called square, and the number m = n -- her in order .

The basic arithmetic operations on matrices are multiplying a matrix by a number, adding and multiplying matrices.

Let's move on to defining the basic operations on matrices.

Matrix addition: The sum of two matrices, for example: A And B , having the same number of rows and columns, in other words, the same orders m And n called matrix C = ( WITH ij )( i = 1, 2, …m; j = 1, 2, …n) the same orders m And n , elements Cij which are equal.

Cij = Aij + Bij (i = 1, 2, …, m; j = 1, 2, …, n) (1.2 )

To denote the sum of two matrices, the notation is used C = A + B. The operation of summing matrices is called their addition

So by definition we have:

+ =

From the definition of the sum of matrices, or more precisely from the formula ( 1.2 ) it immediately follows that the operation of adding matrices has the same properties as the operation of adding real numbers, namely:

1) commutative property: A + B = B + A

2) combining property: (A + B) + C = A + (B + C)

These properties make it possible not to worry about the order of the matrix terms when adding two or more matrices.

Multiplying a matrix by a number :

Matrix product for a real number is called a matrix C = (Cij) (i = 1, 2, … , m; j = 1, 2, …, n) , whose elements are equal

Cij = Aij (i = 1, 2, …, m; j = 1, 2, …, n). (1.3 )

Directly from formula ( 1.3 ) it is clear that multiplying a matrix by a number has the following properties:

1) distributive property regarding the sum of matrices:

( A + B) = A+ B

2) associative property regarding a numerical factor:

() A= ( A)

3) distributive property regarding the sum of numbers:

( + ) A= A + A .

Comment :Difference of two matrices A And B of identical orders it is natural to call such a matrix C of the same orders, which in sum with the matrix B gives the matrix A . To denote the difference between two matrices, a natural notation is used: C = A – B.

Matrix multiplication :

Matrix product A = (Aij) (i = 1, 2, …, m; j = 1, 2, …, n) , having orders respectively equal m And n , per matrix B = (Bij) (i = 1, 2, …, n;

Cij = (i = 1, 2, …, m; j = 1, 2, …, p) (1.4 )

To denote the product of a matrix A to the matrix B use recording

C=AB . The operation of composing a matrix product A to the matrix B called multiplication these matrices. From the above definition it follows that matrix A cannot be multiplied by any matrix B : it is necessary that the number of matrix columns A was equals number of matrix rows B . In order for both works AB And B.A. were not only defined, but also had the same order, it is necessary and sufficient that both matrices A And B were square matrices of the same order.

Formula ( 1.4 ) is a rule for composing matrix elements C ,

which is the product of the matrix A to the matrix B . This rule can be formulated verbally: Element Cij , standing at the intersection i th line and j- th matrix column C=AB , is equal the sum of pairwise products of the corresponding elements i th line matrices A And j- th matrix column B . As an example of the application of this rule, we present the formula for multiplying square matrices of the second order

From the formula ( 1.4 ) the following properties of the matrix product follow: A to the matrix B :

1) associative property: ( AB) C = A(BC);

2) distributive property with respect to the sum of matrices:

(A + B) C = AC + BC or A (B + C) = AB + AC.

It makes sense to raise the question of the permutation property of a product of matrices only for square matrices of the same order. Elementary examples show that the product of two square matrices of the same order does not, generally speaking, have the commutation property. In fact, if we put

A = , B = , That AB = , A BA =

The same matrices for which the product has the commutation property are usually called commuting.

Among square matrices, we highlight the class of so-called diagonal matrices, each of which has elements located outside the main diagonal equal to zero. Among all diagonal matrices with coinciding elements on the main diagonal, two matrices play a particularly important role. The first of these matrices is obtained when all elements of the main diagonal are equal to one, called the identity matrix n- E . The second matrix is obtained with all elements equal to zero and is called the zero matrix n- order and is denoted by the symbol O . Let us assume that there is an arbitrary matrix A , Then

AE=EA=A , AO=OA=O .

The first of the formulas characterizes the special role of the identity matrix E, similar to the role played by the number 1 when multiplying real numbers. As for the special role of the zero matrix ABOUT, then it is revealed not only by the second of the formulas, but also by an elementary verifiable equality: A+O=O+A=A . The concept of a zero matrix can be introduced not for square matrices.

Let's consider any four numbers written in the form of a matrix of two in rows and two columns , Determinant or determinant, made up of the numbers in this table, is the number ad-bc , denoted as follows: .Such a determinant is called second order determinant, since a table of two rows and two columns was taken to compile it. The numbers that make up the determinant are called its elements; at the same time they say that the elements a And d make up main diagonal determinant, and the elements b And c his side diagonal. It can be seen that the determinant is equal to the difference of the products of pairs of elements located on its main and secondary diagonals. The determinant of the third and any other order is approximately the same, namely: Let's say we have a square matrix . The determinant of the following matrix is the following expression: a11a22a33 + a12a23a31 + a13a21a32 – a11a23a32 – a12a21a33 – a13a22a31. . As you can see, it is calculated quite easily if you remember a certain sequence. With a positive sign are the main diagonal and the triangles formed from the elements, which have a side parallel to the main diagonal, in this case these are triangles a12a23a31, a13a21a32 .

There is an easier way to calculate the determinant n- oh order, where n2 . Let's agree to call any element a minor Aij matrices n- first-order determinant corresponding to the matrix that is obtained from the matrix as a result of deleting i th line and j- th column (that row and that column at the intersection of which there is an element Aij ). Element minor Aij will be denoted by the symbol . In this notation, the upper index denotes the row number, the lower index the column number, and the bar over M means that the specified row and column are crossed out. Determinant of order n , corresponding to the matrix, we call the number equal to and denoted by the symbol .

Theorem 1.1 Whatever the line number i ( i =1, 2…, n) , for the determinant n- the first order of magnitude formula is valid

= det A =

called i- th line . We emphasize that in this formula the exponent to which the number is raised (-1) is equal to the sum of the row and column numbers at the intersection of which the element is located Aij .

Theorem 1.2 Whatever the column number j ( j =1, 2…, n) , for the determinant n the th order formula is valid

= det A =

called expansion of this determinant in j- th column .

Determinants also have properties that make the task of calculating them easier. So, below we establish a number of properties that an arbitrary determinant has n -th order.

1. Row-column equality property . Transposition of any matrix or determinant is an operation that results in the rows and columns being swapped while preserving their order. As a result of matrix transposition A the resulting matrix is called a matrix, called transposed with respect to the matrix A and is indicated by the symbol A .

The first property of the determinant is formulated as follows: when transposing, the value of the determinant is preserved, i.e. = .

2. Antisymmetry property when rearranging two rows (or two columns). When two rows (or two columns) are swapped, the determinant retains its absolute value, but changes sign to the opposite. For a second-order determinant, this property can be verified in an elementary way (from the formula for calculating the second-order determinant it immediately follows that the determinants differ only in sign).

3. Linear property of the determinant. We will say that some string ( a) is a linear combination of the other two strings ( b And c ) with coefficients and . The linear property can be formulated as follows: if in the determinant n some order i The th row is a linear combination of two rows with coefficients and , then = + , where

- a determinant that has i The -th row is equal to one of the two rows of the linear combination, and all other rows are the same as , a is the determinant for which i- i string is equal to the second of the two strings, and all other strings are the same as .

These three properties are the main properties of the determinant, revealing its nature. The following five properties are logical consequences three main properties.

Corollary 1. A determinant with two identical rows (or columns) is equal to zero.

Corollary 2. Multiplying all elements of some row (or some column) of a determinant by a number a is equivalent to multiplying the determinant by this number a . In other words, the common factor of all elements of a certain row (or some column) of a determinant can be taken out of the sign of this determinant.

Corollary 3. If all elements of a certain row (or some column) are equal to zero, then the determinant itself is equal to zero.

Corollary 4. If the elements of two rows (or two columns) of a determinant are proportional, then the determinant is equal to zero.

Corollary 5. If to the elements of a certain row (or some column) of a determinant we add the corresponding elements of another row (another column), multiplying by an arbitrary factor , then the value of the determinant does not change. Corollary 5, like the linear property, allows for a more general formulation, which I will give for strings: if to the elements of a certain row of a determinant we add the corresponding elements of a string that is a linear combination of several other rows of this determinant (with any coefficients), then the value of the determinant will not change . Corollary 5 is widely used in the concrete calculation of determinants.

, This system can be written as a matrix as follows: A= , where the answers to the equations will be in the last column. We will now introduce the concept of a fundamental determinant; in this case it will look like this:

= . The main determinant, as you have already noticed, is a matrix made up of the coefficients of the variables. They also appear in column order, i.e. the first column contains the coefficients that are found at x , in the second column at y , and so on. This is very important, because in the following steps we will replace each column of coefficients for a variable with a column of equation answers. So, as I said, we replace the column at the first variable with the answer column, then at the second, of course it all depends on how many variables we need to find.

1 = , 2 = , 3 = .

Then you need to find the determinants 1, 2, 3. You already know how to find the third-order determinant. A This is where we apply Cramer's rule. It looks like this:

x1 = , x2 = , x3 = – for this case, but in general it looks like this: x i = . A determinant made up of coefficients for unknowns is called determinant of the system .

1. V. A. Ilyin, E. G. Poznyak “Linear Algebra”