ODA. A rectangular table consisting of T lines and P columns of real numbers are called matrix size t×p. Matrices are denoted by capital Latin letters: A, B,..., and an array of numbers is separated by round or square brackets.
The numbers included in the table are called matrix elements and are denoted in small Latin letters with a double index, where i– line number, j– number of the column at the intersection of which the element is located. IN general view the matrix is written like this:
Two matrices are considered equal, if their corresponding elements are equal.
If the number of matrix rows T equal to the number of its columns P, then the matrix is called square(otherwise – rectangular).
Size Matrix 
called a row matrix. Size Matrix 
called a column matrix.
Matrix elements having equal indices ( 
etc.), form main diagonal matrices. The other diagonal is called the side diagonal.
The square matrix is called diagonal, if all its elements located outside the main diagonal are equal to zero.
A diagonal matrix whose diagonal elements are equal to one is called single matrix and has the standard notation E:
If all matrix elements located above (or below) the main diagonal are equal to zero, the matrix is said to have a triangular form:
§2. Operations on matrices
1. Matrix transposition - a transformation in which the rows of the matrix are written as columns while maintaining their order. For a square matrix, this transformation is equivalent to a symmetric mapping about the main diagonal:
.
2. Matrices of the same dimension can be summed (subtracted). The sum (difference) of matrices is a matrix of the same dimension, each element of which is equal to the sum (difference) of the corresponding elements of the original matrices:
3. Any matrix can be multiplied by a number. The product of a matrix by a number is a matrix of the same order, each element of which is equal to the product of the corresponding element of the original matrix by this number:
.
4. If the number of columns of one matrix is equal to the number of rows of another, then you can multiply the first matrix by the second. The product of such matrices is a matrix, each element of which is equal to the sum of pairwise products of the elements of the corresponding row of the first matrix and the elements of the corresponding column of the second matrix.
Consequence. Matrix exponentiation To>1 is the product of matrix A To once. Defined only for square matrices.
Example.
Properties of operations on matrices.
(A+B)+C=A+(B+C);
k(A+B)=kA+kV;
A(B+C)=AB+AC;
(A+B)C=AC+BC;
k(AB)=(kA)B=A(kV);
A(BC)=(AB)C;
(kA) T = kA T;
(A+B) T =A T +B T;
(AB) T =B T A T;
The properties listed above are similar to the properties of operations on numbers. There are also specific properties of matrices. These include, for example, the distinctive property of matrix multiplication. If the product AB exists, then the product BA
May not exist
May differ from AB.
Example. The company produces products of two types A and B and uses three types of raw materials S 1, S 2, and S 3. Raw material consumption rates are specified by the matrix N= 
, Where n ij– quantity of raw materials j, spent on the production of a unit of output i. The production plan is given by the matrix C=(100 200), and the unit cost of each type of raw material is given by the matrix 
. Determine the raw material costs required for planned production and the total cost of raw materials.
Solution. We define raw material costs as the product of matrices C and N:
We calculate the total cost of raw materials as the product of S and P.
>> Matrices
4.1.Matrixes. Operations on matrices
A rectangular matrix of size mxn is a collection of mxn numbers arranged in the form of a rectangular table containing m rows and n columns. We will write it in the form
or abbreviated as A = (a i j) (i = ; j = ), numbers a i j are called its elements; The first index indicates the row number, the second - the column number. A = (a i j) and B = (b i j) same size are called equal if their elements standing in the same places are pairwise equal, that is, A = B if a i j = b i j .
A matrix consisting of one row or one column is called a row vector or a column vector, respectively. Column vectors and row vectors are simply called vectors.
A matrix consisting of one number is identified with this number. A of size mxn, all elements of which are equal to zero, are called zero and are denoted by 0. Elements with the same indices are called elements of the main diagonal. If the number of rows is equal to the number of columns, that is, m = n, then the matrix is called a square matrix of order n. Square matrices in which only the elements of the main diagonal are nonzero are called diagonal and are written as follows:
.
If all elements a i i of the diagonal are equal to 1, then it is called unit and is denoted by the letter E:
.
A square matrix is called triangular if all elements above (or below) the main diagonal are equal to zero. Transposition is a transformation in which rows and columns are swapped while maintaining their numbers. Transposition is indicated by a T at the top.
If we rearrange the rows and columns in (4.1), we get
,
which will be transposed with respect to A. In particular, when transposing a column vector, a row vector is obtained and vice versa.
The product of A and the number b is a matrix whose elements are obtained from the corresponding elements of A by multiplying by the number b: b A = (b a i j).
The sum A = (a i j) and B = (b i j) of the same size is called C = (c i j) of the same size, the elements of which are determined by the formula c i j = a i j + b i j.
The product AB is determined under the assumption that the number of columns of A is equal to the number of rows of B.
The product AB, where A = (a i j) and B = (b j k), where i = , j= , k= , specified in in a certain order AB is called C = (c i k), the elements of which are determined by next rule:
c i k = a i 1 b 1 k + a i 2 b 2 k +... + a i m b m k = a i s b s k . (4.2)
In other words, the element of the product AB is defined as follows: element i-th line and the kth column C is equal to the sum of the products elements of the i-th rows A to the corresponding elements of the kth column B.
Example 2.1. Find the product of AB and .
Solution. We have: A of size 2x3, B of size 3x3, then the product AB = C exists and the elements of C are equal
From 11 = 1×1 +2×2 + 1×3 = 8, from 21 = 3×1 + 1×2 + 0×3 = 5, from 12 = 1×2 + 2×0 + 1×5 = 7 ,
s 22 = 3×2 + 1 × 0 + 0 × 5 = 6, s 13 = 1 × 3 + 2 × 1 + 1 × 4 = 9, s 23 = 3 × 3 + 1 × 1 + 0 × 4 = 10 .
, and the product BA does not exist.
Example 2.2. The table shows the number of units of products shipped daily from dairies 1 and 2 to stores M 1, M 2 and M 3, and delivery of a unit of product from each dairy to store M 1 costs 50 den. units, to the M 2 store - 70, and to M 3 - 130 den. units Calculate the daily transportation costs of each plant.
| Dairy plant |
|||
Solution. Let us denote by A the matrix given to us in the condition, and by
B - matrix characterizing the cost of delivering a unit of product to stores, i.e.,
,
Then the transportation cost matrix will look like:
.
So, the first plant spends 4,750 deniers on transportation daily. units, the second - 3680 monetary units.
Example 2.3. The sewing company produces winter coats, demi-season coats and raincoats. The planned output for a decade is characterized by the vector X = (10, 15, 23). Four types of fabrics are used: T 1, T 2, T 3, T 4. The table shows the fabric consumption rates (in meters) for each product. Vector C = (40, 35, 24, 16) specifies the cost of a meter of fabric of each type, and vector P = (5, 3, 2, 2) specifies the cost of transporting a meter of fabric of each type.
| Fabric consumption |
||||
| Demi-season coat |
||||
1. How many meters of each type of fabric will be needed to complete the plan?
2. Find the cost of fabric spent on sewing each type of product.
3. Determine the cost of all the fabric needed to complete the plan.
Solution. Let us denote by A the matrix given to us in the condition, i.e.,
,
then to find the number of meters of fabric needed to complete the plan, you need to multiply vector X by matrix A:
We find the cost of fabric spent on sewing products of each type by multiplying matrix A and vector C T:
.
The cost of all the fabric needed to complete the plan will be determined by the formula:
Finally, taking into account transport costs, the entire amount will be equal to the cost of the fabric, i.e. 9472 den. units, plus value
X A P T = 
.
So, X A C T + X A P T = 9472 + 1037 = 10509 (money units).
Let there be a square matrix of nth order
Matrix A -1 is called inverse matrix in relation to matrix A, if A*A -1 = E, where E is the identity matrix of the nth order.
Identity matrix- such a square matrix in which all elements are along the main diagonal, running from the upper left corner to the right bottom corner, are ones, and the rest are zeros, for example:
inverse matrix may exist only for square matrices those. for those matrices in which the number of rows and columns coincide.
Theorem for the existence condition of an inverse matrix
In order for a matrix to have an inverse matrix, it is necessary and sufficient that it be non-singular.
The matrix A = (A1, A2,...A n) is called non-degenerate, if the column vectors are linearly independent. The number of linearly independent column vectors of a matrix is called the rank of the matrix. Therefore, we can say that in order for there to exist inverse matrix, it is necessary and sufficient that the rank of the matrix is equal to its dimension, i.e. r = n.
Algorithm for finding the inverse matrix
- Write matrix A into the table for solving systems of equations using the Gaussian method and assign matrix E to it on the right (in place of the right-hand sides of the equations).
- Using Jordan transformations, reduce matrix A to a matrix consisting of unit columns; in this case, it is necessary to simultaneously transform the matrix E.
- If necessary, rearrange the rows (equations) of the last table so that under the matrix A of the original table you get the identity matrix E.
- Write down the inverse matrix A -1, which is located in the last table under the matrix E of the original table.
For matrix A, find the inverse matrix A -1
Solution: We write down matrix A and assign to the right identity matrix E. Using Jordan transformations, we reduce matrix A to the identity matrix E. Calculations are given in table 31.1.
Let's check the correctness of the calculations by multiplying the original matrix A and the inverse matrix A -1.
As a result of matrix multiplication, the identity matrix was obtained. Therefore, the calculations were made correctly.
Answer: 
Solving matrix equations
Matrix equations can look like:
AX = B, HA = B, AXB = C,
where A, B, C are the specified matrices, X is the desired matrix.
Matrix equations are solved by multiplying the equation by inverse matrices.
For example, to find the matrix from the equation, you need to multiply this equation by on the left.
Therefore, to find a solution to the equation, you need to find the inverse matrix and multiply it by the matrix on the right side of the equation.
Other equations are solved similarly.
Solve the equation AX = B if 
Solution: Since the inverse matrix is equal to (see example 1)
Matrix method in economic analysis
Along with others, they are also used matrix methods. These methods are based on linear and vector-matrix algebra. Such methods are used for the purposes of analyzing complex and multidimensional economic phenomena. Most often these methods are used when necessary comparative assessment functioning of organizations and their structural divisions.
In the process of applying matrix analysis methods, several stages can be distinguished.
At the first stage the system is being formed economic indicators and on its basis, a source data matrix is compiled, which is a table in which system numbers are shown in its individual rows (i = 1,2,....,n), and in vertical columns - numbers of indicators (j = 1,2,....,m).
At the second stage For each vertical column, the largest of the available indicator values is identified, which is taken as one.
After this, all amounts reflected in this column are divided by highest value and a matrix of standardized coefficients is formed.
At the third stage all components of the matrix are squared. If they have different significance, then each matrix indicator is assigned a certain weight coefficient k. The value of the latter is determined by expert opinion.
On the last one, fourth stage found rating values Rj are grouped in order of their increase or decrease.
The matrix methods outlined should be used, for example, when comparative analysis various investment projects, as well as when assessing other economic indicators of organizations.
Definition 1. Matrix A sizem n is a rectangular table of m rows and n columns, consisting of numbers or other mathematical expressions (called matrix elements), i = 1,2,3,…,m, j = 1,2,3,…,n.
, or
Definition 2.
Two matrices 
And 
same size are called equal, if they coincide element by element, i.e. 
=
,i = 1,2,3,…,m, j = 1,2,3,…,n.
Using matrices, it is easy to record some economic dependencies, for example, tables of resource distribution for certain sectors of the economy.
Definition 3. If the number of rows of a matrix coincides with the number of its columns, i.e. m = n, then the matrix is called square ordern, otherwise rectangular.
Definition 4. The transition from matrix A to matrix A m, in which the rows and columns are swapped while maintaining order, is called transposition matrices.
Types of matrices: square (size 33) - 
,
rectangular (size 25) - 
,
diagonal - 
, single - 
, zero - 
,
matrix-row - 
, matrix-column -.
Definition 5.
Elements of a square matrix of order n with the same indices are called elements of the main diagonal, i.e. these are the elements: 
.
Definition 6. Elements of a square matrix of order n are called elements of the secondary diagonal if the sum of their indices is equal to n + 1, i.e. these are the elements: .
1.2. Operations on matrices.
1
0
.
Amount
two matrices 
And 
of the same size is called a matrix C = (with ij), the elements of which are determined by the equality with ij = a ij + b ij, (i = 1,2,3,…,m, j = 1,2,3,…,n).
Properties of the matrix addition operation.
For any matrices A, B, C of the same size, the following equalities hold:
1) A + B = B + A (commutativity),
2) (A + B) + C = A + (B + C) = A + B + C (associativity).
2
0
.
The work
matrices
per number
called a matrix 
the same size as matrix A, and b ij = 
(i = 1,2,3,…,m, j = 1,2,3,…,n).
Properties of the operation of multiplying a matrix by a number.
(A) = ()A (associativity of multiplication);
(A+B) = A+B (distributivity of multiplication relative to matrix addition);
(+)A = A+A (distributivity of multiplication relative to the addition of numbers).
Definition 7.
Linear combination of matrices 
And 
of the same size is called an expression of the form A+B, where and are arbitrary numbers.
3 0 . Product A In matrices A and B, respectively, of size mn and nk, is called a matrix C of size mk, such that the element with ij is equal to the sum of the products of the elements of the i-th row of matrix A and the j-th column of matrix B, i.e. with ij = a i 1 b 1 j +a i 2 b 2 j +…+a ik b kj .
The product AB exists only if the number of columns of matrix A coincides with the number of rows of matrix B.
Properties of the matrix multiplication operation:
(AB)C = A(BC) (associativity);
(A+B)C = AC+BC (distributivity with respect to matrix addition);
A(B+C) = AB+AC (distributivity with respect to matrix addition);
AB BA (not commutative).
Definition 8. Matrices A and B, for which AB = BA, are called commuting or commuting.
Multiplying a square matrix of any order by the corresponding identity matrix does not change the matrix.
Definition 9. Elementary transformations The following operations are called matrices:
Swap two rows (columns).
Multiplying each element of a row (column) by a number other than zero.
Adding to the elements of one row (column) the corresponding elements of another row (column).
Definition 10. Matrix B obtained from matrix A using elementary transformations is called equivalent(denoted by BA).
Example 1.1. Find a linear combination of matrices 2A–3B if
,
.
,
,
.
Example
1.2.
Find the product of matrices 
, If
.
Solution: since the number of columns of the first matrix coincides with the number of rows of the second matrix, then the product of matrices exists. As a result, we obtain a new matrix 
, Where
As a result we get 
.
Lecture 2. Determinants. Calculation of second and third order determinants. Properties of determinantsn-th order.
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 the rows and columns in it 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 a 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:
