Definition by the Matrix– called a table of numbers containing a certain number of rows and columns
The elements of the matrix are numbers of the form a ij, where i is the row number j is the column number
Example 1 i = 2 j = 3
Designation: A=
Types of matrices:
1. If the number of rows is not equal to the number of columns, then the matrix is called rectangular:
2. If the number of rows is equal to the number of columns, then the matrix is called square:
Number of rows or columns square matrix called her in order. In the example n = 2
Consider a square matrix of order n:
The diagonal containing the elements a 11, a 22......., a nn is called main , and the diagonal containing the elements a 12, a 2 n -1, …….a n 1 – auxiliary.
A matrix in which only the elements on the main diagonal are nonzero is called diagonal:
Example 4 
n=3
3. If a diagonal matrix has elements equal to 1, then the matrix is called single and is designated by the letter E:
Example 6 
n=3
4. A matrix, all elements of which are equal to zero, is called null matrix and is denoted by the letter O
Example 7 
5. Triangular A matrix of nth order is a square matrix, all of whose elements located below the main diagonal are equal to zero:
Example 8 
n=3
Actions on matrices:
The sum of a matrix A and B is a matrix C whose elements are equal to the sum of the corresponding elements of matrices A and B.
Only matrices that have same number rows and columns.
Product of matrix A and number k such a matrix kA is called, each element of which is equal to ka ij
Example10 
Multiplying a matrix by a number is reduced to multiplying all the elements of the matrix by that number.
Product of matrices To multiply a matrix by a matrix, you need to select the first row of the first matrix and multiply by the corresponding elements of the first column of the second matrix, and add the result. Place this result in the result matrix in the 1st row and 10th column. We perform the same actions with all other elements: 1st line to the second column, to the 3rd, etc., then with the following lines.
Example 11 
Multiplying matrix A by matrix B is possible only if the number of columns of the first matrix is equal to the number of columns of the second matrix.
- the work exists;
- the work does not exist
Examples 12 
there is nothing to multiply the last line in matrix II with, i.e. the work does not exist
Matrix Transpose The operation of replacing row elements with column elements is called:
Example13 
By raising to a power is called sequential multiplication of a matrix by itself.
Note that matrix elements can be not only numbers. Let's imagine that you are describing the books that are on your bookshelf. Let your shelf be in order and all books be in strictly defined places. The table, which will contain a description of your library (by shelves and the order of books on the shelf), will also be a matrix. But such a matrix will not be numeric. Another example. Instead of numbers there are different functions, united by some dependence. The resulting table will also be called a matrix. In other words, the Matrix is any rectangular table, composed of homogeneous elements. Here and further we will talk about matrices made up of numbers.
Instead of parentheses used to write matrices square brackets or straight double vertical lines
 |
(2.1*) |
Definition 2. If in the expression(1) m = n, then they talk about square matrix, what if , then oh rectangular.
Depending on the values of m and n, there are some special types matrices:
The most important characteristic square matrix is her determinant or determinant, which is made up of matrix elements and is denoted
Obviously, D E =1; .
Definition 3. If , then the matrix A called non-degenerate or not special.
Definition 4. If detA = 0 , then the matrix A called degenerate or special.
Definition 5. Two matrices A And B are called equal and write A = B if they have same sizes and their corresponding elements are equal, i.e..
For example, matrices and are equal, because they are equal in size and each element of one matrix is equal to the corresponding element of the other matrix. But the matrices cannot be called equal, although the determinants of both matrices are equal, and the sizes of the matrices are the same, but not all elements located in the same places are equal. The matrices are different because they have different size. The first matrix is 2x3 in size, and the second is 3x2. Although the number of elements is the same - 6 and the elements themselves are the same 1, 2, 3, 4, 5, 6, but they are in different places in each matrix. But the matrices are equal, according to Definition 5.
Definition 6. If you fix a certain number of matrix columns A and the same number of rows, then the elements at the intersection of the indicated columns and rows form a square matrix n- th order, the determinant of which called minor k – th order matrix A.
Example. Write down three second-order minors of the matrix
Matrix
dimension is a table of numbers containing rows and columns. The numbers are called the elements of this matrix, where is the row number, is the column number, at the intersection of which is this element. A matrix containing rows and columns has the form: 
.
Types of matrices:
1) at – square , and they call matrix order ;
2) a square matrix in which all non-diagonal elements are equal to zero
– diagonal ;
3) a diagonal matrix in which all diagonal elements are equal
unit – single and is denoted by ;
4) at – rectangular ;
5) when – row matrix (row vector);
6) when – matrix-column (vector-column);
7) for all – zero matrix.
Note that the main numerical characteristic of a square matrix is its determinant. The determinant corresponding to a matrix of the th order also has the th order.
Determinant of a 1st order matrix called number.
Determinant of a 2nd order matrix
called number 
. (1.1)
Determinant of a 3rd order matrix
called number 
. (1.2)
Let us present the definitions necessary for further presentation.
Minor M ij element A ij matrices n- order A is called the determinant of the matrix ( n-1)- th order obtained from matrix A by deleting i-th line and j th column.
Algebraic complement A ij element A ij matrices n- of order A is the minor of this element, taken with the sign .
Let us formulate the basic properties of determinants that are inherent in determinants of all orders and simplify their calculation.
1. When a matrix is transposed, its determinant does not change.
2. When rearranging two rows (columns) of a matrix, its determinant changes sign.
3. A determinant that has two proportional (equal) rows (columns) is equal to zero.
4. Total multiplier elements of any row (column) of the determinant can be taken out of the sign of the determinant.
5. If the elements of any row (column) of a determinant represent the sum of two terms, then the determinant can be decomposed into the sum of two corresponding determinants.
6. The determinant will not change if the corresponding elements of its other row (column), previously multiplied by any number, are added to the elements of any of its rows (columns).
7. Matrix determinant equal to the sum products of the elements of any of its rows (columns) by algebraic additions these elements.
Let's explain this property using the example of a 3rd order determinant. IN in this case property 7 means that – decomposition of the determinant into elements of the 1st row. Note that for decomposition, select the row (column) where there is zero elements, since the terms corresponding to them in the expansion vanish.
Property 7 is a determinant decomposition theorem formulated by Laplace.
8. The sum of the products of the elements of any row (column) of a determinant by the algebraic complements of the corresponding elements of its other row (column) is equal to zero.
The last property is often called pseudo-decomposition of the determinant.
Self-test questions.
1. What is called a matrix?
2. What matrix is called square? What is meant by its order?
3. What matrix is called diagonal, identity?
4. Which matrix is called a row matrix and a column matrix?
5. What is the main numerical characteristic of a square matrix?
6. What number is called the determinant of the 1st, 2nd and 3rd order?
7. What is called the minor and algebraic complement of a matrix element?
8. What are the main properties of determinants?
9. Using what property can one calculate the determinant of any order?
Actions on matrices(scheme 2)
A number of operations are defined on a set of matrices, the main ones being the following:
1) transposition – replacing matrix rows with columns, and columns with rows;
2) multiplying a matrix by a number is done element-by-element, that is 
, Where ,
;
3) matrix addition, defined only for matrices of the same dimension;
4) multiplication of two matrices, defined only for matched matrices.
The sum (difference) of two matrices such a resulting matrix is called, each element of which is equal to the sum (difference) of the corresponding elements of the matrix-commands.
The two matrices are called agreed upon
, if the number of columns of the first one is equal to the number of rows of the other. Product of two matched matrices
and such a resulting matrix is called 
, What , (1.4)
Where , . It follows that the element of the th row and the th column of the matrix is equal to the sum of pairwise products of the elements of the th row of the matrix and the elements of the th column of the matrix.
The product of matrices is not commutative, that is, A . B B . A. An exception is, for example, the product of square matrices and unit A . E = E . A.
Example 1.1. Multiply matrices A and B if:
.
Solution. Since the matrices are consistent (the number of matrix columns is equal to the number of matrix rows), we will use formula (1.4):
Self-test questions.
1. What actions are performed on matrices?
2. What is called the sum (difference) of two matrices?
3. What is called the product of two matrices?
Cramer's method for solving quadratic linear systems algebraic equations (scheme 3)
Let us give a number of necessary definitions.
System linear equations called heterogeneous , if at least one of its free terms is different from zero, and homogeneous , if all its free terms are equal to zero.
Solving a system of equations is an ordered set of numbers that, when substituted for variables in a system, turns each of its equations into an identity.
The system of equations is called joint , if it has at least one solution, and non-joint , if she has no solutions.
The simultaneous system of equations is called certain if she has the only solution, And uncertain , if it has more than one solution.
Let us consider an inhomogeneous square system linear algebraic equations, having the following general form:
. (1.5) The main matrix of the system
linear algebraic equations is a matrix composed of coefficients associated with the unknowns: 
.
The determinant of the main matrix of the system is called main determinant and is designated .
The auxiliary determinant is obtained from the main determinant by replacing the th column with a column of free terms.
Theorem 1.1 (Cramer's theorem). If the main determinant of a quadratic system of linear algebraic equations is nonzero, then the system has a unique solution, calculated by the formulas:
If the main determinant is , then the system either has an infinite number of solutions (for all zero auxiliary determinants) or has no solution at all (if at least one of the auxiliary determinants differs from zero)
In light of the above definitions, Cramer's theorem can be formulated differently: if the main determinant of a system of linear algebraic equations is nonzero, then the system is jointly defined and at the same time ; if the main determinant is zero, then the system is either jointly indefinite (for all ) or inconsistent (if at least one of them differs from zero).
After this, the resulting solution should be checked.
Example 1.2. Solve the system using Cramer's method 
Solution. Since the main determinant of the system
is different from zero, then the system has a unique solution. Let's calculate auxiliary determinants
Let's use Cramer's formulas (1.6): 
, ,
Self-test questions.
1. What is called solving a system of equations?
2. Which system of equations is called compatible or incompatible?
3. Which system of equations is called definite or indefinite?
4. Which matrix of the system of equations is called the main one?
5. How to calculate auxiliary determinants of a system of linear algebraic equations?
6. What is the essence of Cramer’s method for solving systems of linear algebraic equations?
7. What can a system of linear algebraic equations be like if its main determinant is zero?
Solving quadratic systems of linear algebraic equations using the method inverse matrix (scheme 4)
A matrix having a nonzero determinant is called non-degenerate ; having a determinant equal to zero – degenerate .
The matrix is called the inverse for a given square matrix, if when multiplying the matrix by its inverse both on the right and on the left, the identity matrix is obtained, that is. (1.7)
Note that in this case the product of matrices and is commutative.
Theorem 1.2. Necessary and sufficient condition the existence of an inverse matrix for a given square matrix is the non-zero determinant of the given matrix
If the main matrix of the system turns out to be singular during testing, then there is no inverse for it, and the method under consideration cannot be applied.
If the main matrix is non-singular, that is, the determinant is 0, then the inverse matrix can be found for it using the following algorithm.
1. Calculate the algebraic complements of all matrix elements.
2. Write the found algebraic additions into the matrix transposed.
3. Create an inverse matrix using the formula: 
(1.8)
4. Check the correctness of the found matrix A-1 according to formula (1.7). Note that this check can be included in the final check of the system solution itself.
System (1.5) of linear algebraic equations can be represented as a matrix equation: , where is the main matrix of the system, is the column of unknowns, and is the column of free terms. Let's multiply this equation on the left by the inverse matrix, we get:
Since, by definition of the inverse matrix, the equation takes the form 
or . (1.9)
Thus, to solve a quadratic system of linear algebraic equations, you need to multiply the column of free terms on the left by the matrix inverse of the main matrix of the system. After this, you should check the resulting solution.
Example 1.3. Solve the system using the inverse matrix method
Solution. Let us calculate the main determinant of the system
. Consequently, the matrix is non-singular and its inverse matrix exists.
Let's find the algebraic complements of all elements of the main matrix:
Let us write the algebraic additions transposed into the matrix
. Let us use formulas (1.8) and (1.9) to find a solution to the system
Self-test questions.
1. Which matrix is called singular, non-degenerate?
2. What matrix is called the inverse of a given one? What is the condition of its existence?
3. What is the algorithm for finding the inverse matrix for a given one?
4. Which one matrix equation is a system of linear algebraic equations equivalent?
5. How to solve a system of linear algebraic equations using the inverse matrix for the main matrix of the system?
Study of inhomogeneous systems of linear algebraic equations(scheme 5)
The study of any system of linear algebraic equations begins with the transformation of its extended matrix by the Gaussian method. Let the dimension of the main matrix of the system be equal to .
Matrix called extended matrix of the system , if, along with the coefficients of the unknowns, it contains a column of free terms. Therefore, the dimension is .
The Gaussian method is based on elementary transformations , which include:
– rearrangement of matrix rows;
– multiplying the rows of the matrix by a number different from the steering wheel;
– element-wise addition of matrix rows;
– deleting the zero line;
– matrix transposition (in this case, transformations are performed column-wise).
Elementary transformations lead the original system to a system equivalent to it. Systems are called equivalent , if they have the same set of solutions.
Matrix rank called highest order its minors that are nonzero. Elementary transformations do not change the rank of the matrix.
When asked about the availability of solutions homogeneous system linear equations are answered by the following theorem.
Theorem 1.3 (Kronecker-Capelli theorem). A non-homogeneous system of linear algebraic equations is consistent if and only if the rank of the extended matrix of the system is equal to the rank of its main matrix, i.e. 
Let us denote the number of rows remaining in the matrix after the Gaussian method by (accordingly, the number of equations remaining in the system). These lines matrices are called basic .
If , then the system has a unique solution (is jointly defined), its matrix is reduced to a triangular form by elementary transformations. Such a system can be solved using the Cramer method, using the inverse matrix, or the universal Gauss method.
If (the number of variables in the system is greater than equations), the matrix is reduced by elementary transformations to stepped view. Such a system has many solutions and is jointly uncertain. In this case, to find solutions to the system, it is necessary to perform a number of operations.
1. Leave the system of unknowns on the left sides of the equations ( basic variables ), the rest of the unknowns are moved to the right sides ( free variables ). After dividing the variables into basic and free system takes the form:
. (1.10)
2. From the coefficients of the basic variables, make up a minor ( basic minor ), which must be non-zero.
3. If the basic minor of system (1.10) is equal to zero, then replace one of the basic variables with a free one; Check the resulting basis minor for non-zero.
4. Applying formulas (1.6) of the Cramer method, counting the right-hand sides of their equations free members, find the expression of the basic variables through the free ones in general view. The resulting ordered set of system variables is its general decision .
5. Giving the free variables in (1.10) arbitrary values, calculate the corresponding values of the basic variables. The resulting ordered set of values of all variables is called private solution systems corresponding to given values of free variables. The system has an infinite number of particular solutions.
6. Get basic solution system – a particular solution obtained for zero values of free variables.
Note that the number of basis sets of variables of system (1.10) is equal to the number of combinations of elements by elements. Since each basic set of variables has its own basic solution, therefore, the system also has basic solutions.
A homogeneous system of equations is always consistent, since it has at least one – zero (trivial) solution. In order for a homogeneous system of linear equations with variables to have non-zero solutions, it is necessary and sufficient that its main determinant be equal to zero. This means that the rank of its main matrix less number unknown In this case, the study of a homogeneous system of equations for general and particular solutions is carried out similarly to the study of a non-homogeneous system. Solutions of a homogeneous system of equations have important property: if two are known various solutions homogeneous system of linear equations, then their linear combination is also a solution to this system. It is easy to verify the validity of the following theorem.
Theorem 1.4. The general solution of an inhomogeneous system of equations is the sum of the general solution of the corresponding homogeneous system and some particular solution of the inhomogeneous system of equations
Example 1.4.
Explore the given system and find one particular solution:
Solution. Let us write down the extended matrix of the system and apply it to it elementary transformations:
. Since and , then by Theorem 1.3 (Kronecker-Capelli) given system linear algebraic equations are consistent. The number of variables, i.e., means the system is uncertain. The number of basis sets of system variables is equal to
. Consequently, 6 sets of variables can be basic: . Let's consider one of them. Then the system obtained as a result of the Gauss method can be rewritten in the form
. Main determinant 
. Using Cramer's method, we look for a general solution to the system. Auxiliary qualifiers
According to formulas (1.6) we have
. This expression basis variables through free ones represents the general solution of the system:
At specific values free variables, from the general solution we obtain a particular solution of the system. For example, a private solution 
corresponds to the values of free variables 
. At we obtain the basic solution of the system 
Self-test questions.
1. Which system of equations is called homogeneous or inhomogeneous?
2. Which matrix is called extended?
3. List the basic elementary transformations of matrices. What method of solving systems of linear equations is based on these transformations?
4. What is the rank of a matrix? How can you calculate it?
5. What does the Kronecker-Capelli theorem say?
6. To what form can a system of linear algebraic equations be reduced as a result of its solution by the Gauss method? What does this mean?
7. Which rows of the matrix are called basic?
8. What system variables are called basic, which ones are free?
9. What solution of an inhomogeneous system is called private?
10.Which of its solutions is called basic? How many basic solutions does an inhomogeneous system of linear equations have?
11.What solution of an inhomogeneous system of linear algebraic equations is called general? Formulate a theorem about general decision inhomogeneous system of equations.
12. What are the main properties of solutions to a homogeneous system of linear algebraic equations?
Various operations are performed on such matrices: they multiply by each other, find determinants, etc. Matrix - special case array: if an array can have any number of dimensions, then only a two-dimensional array is called a matrix.
In programming, a matrix is also called a two-dimensional array. Any of the arrays in the program has a name, as if it were a single variable. To clarify which of the array cells is meant, when it is mentioned in the program, the number of the cell in it is used together with the variable. Both a two-dimensional matrix and an n-dimensional array in a program can contain not only numeric, but also symbolic, string, Boolean and other information, but always the same within the entire array.
Matrices are designated in capital letters A:MxN, where A is the name of the matrix, M is the number of rows in the matrix, and N is the number of columns. Elements – corresponding lowercase letters with indices indicating their number in row and column a (m, n).
The most common matrices are rectangular in shape, although in the distant past mathematicians also considered triangular ones. If the number of rows and columns of a matrix is the same, it is called square. In this case, M=N already has the name of the matrix order. A matrix with only one row is called a row. A matrix with only one column is called a columnar matrix. A diagonal matrix is a square matrix in which only the elements located along the diagonal are non-zero. If all elements are equal to one, the matrix is called identity; if all elements are equal to zero, it is called zero.
If you swap rows and columns in a matrix, it becomes transposed. If all elements are replaced by complex conjugates, it becomes complex conjugate. In addition, there are other types of matrices, determined by the conditions that are imposed on the matrix elements. But most of these conditions apply only to square ones.
Video on the topic
The matrix is denoted by capital Latin letters ( A, IN, WITH,...).
Definition 1. Rectangular table view,
consisting of m lines and n columns is called matrix.
Matrix element, i – row number, j – column number.
Types of matrices:
elements on the main diagonal:
trA=a 11 +a 22 +a 33 +…+a nn .
§2. Determinants of 2nd, 3rd and nth order
Let two square matrices be given:
Definition 1. Determinant of the second order matrix A 1
is a number denoted by ∆ and equal to 
, Where
Example. Calculate the 2nd order determinant:
Definition 2. Determinant of the 3rd order of a square matrix A 2 is called a number of the form:
This is one way to calculate the determinant.
Example. Calculate
Definition 3. If a determinant consists of n-rows and n-columns, then it is called an nth-order determinant.
Properties of determinants:
The determinant does not change when transposed (that is, if its rows and columns are swapped while maintaining the order).
If you swap any two rows or two columns in the determinant, then the determinant will only change the sign.
The common factor of any row (column) can be taken beyond the sign of the determinant.
If all elements of any row (column) of a determinant are equal to zero, then the determinant is equal to zero.
The determinant is zero if the elements of any two rows are equal or proportional.
The determinant will not change if the corresponding elements of another row (column) are added to the elements of one row (column), multiplied by the same number.
Example.
Definition 4. The determinant obtained from a given one by crossing out a column and a row is called minor the corresponding element. M ij element a ij .
Definition 5. Algebraic complement element a ij is called the expression
§3. Actions on matrices
Linear operations
1) When adding matrices, their elements of the same name are added.
When subtracting matrices, their elements of the same name are subtracted.
When multiplying a matrix by a number, each element of the matrix is multiplied by that number:
3.2.Matrix multiplication.
Work matrices A to the matrix IN there is a new matrix whose elements are equal to the sum of the products of the elements of the i-th row of the matrix A to the corresponding elements of the jth column of the matrix IN. Matrix product A to the matrix IN can be found only if the number of matrix columns A equal to the number of rows of the matrix IN. Otherwise, the work is impossible.
Comment:
(does not obey the commutative property)
§ 4. Inverse matrix
The inverse matrix exists only for a square matrix, and the matrix must be non-singular.
Definition 1. Matrix A called non-degenerate, if the determinant of this matrix is not equal to zero
Definition 2. A-1 is called inverse matrix for a given non-singular square matrix A, if when multiplying this matrix by the given one, both on the right and on the left, the identity matrix is obtained.
Algorithm for calculating the inverse matrix
1 way (using algebraic additions)
Example 1:
