Replacing the rows of a matrix with the corresponding columns is called. matrices

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
the same size are called equal, if they match element by element, i.e. =,i = 1,2,3,…,m, j = 1,2,3,…,n.

With the help of matrices, it is easy to write down some economic dependencies, for example, tables of the distribution of resources for certain sectors of the economy.

Definition 3. If the number of matrix rows matches the number of its columns, i.e. m = n, then the matrix is ​​called square ordern, otherwise rectangular.

Definition 4. The transition from a matrix A to a matrix A m, in which the rows and columns are swapped with the preservation of order, is called transposition matrices.

Types of matrices: square (size 33) -
,

rectangular (size 25) -
,

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 secondary diagonal elements if the sum of their indices is equal to n + 1, i.e. these are the elements: .

1.2. Operations on matrices.

1 0 . sum two matrices
and
of the same size is called a matrix С = (с ij), the elements of which are determined by equality with ij = a ij + b ij , (i = 1,2,3,…,m, j = 1,2,3,…,n).

Properties of the operation of matrix addition.

For any matrices A,B,C of the same size, the following equalities are fulfilled:

1) A + B = B + A (commutativity),

2) (A + B) + C = A + (B + C) = A + B + C (associativity).

2 0 . work matrices
per number  called 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.

(А) = ()А (associativity of multiplication);

(А+В) = А+В (distributivity of multiplication with respect to matrix addition);

(+)A = A+A (distributivity of multiplication with respect to 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 sizes mn and nk, is called a matrix C of size mk, 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 is the same as the number of rows of matrix B.

Properties of the operation of matrix multiplication:

(АВ)С = А(ВС) (associativity);

(А+В)С = АС+ВС (distributivity with respect to matrix addition);

А(В+С) = АВ+АС (distributivity with respect to matrix addition);

АВ  ВА (not commutativity).

Definition 8. Matrices A and B, for which AB = BA, are called commuting or permuting.

Multiplying a square matrix of any order by the corresponding identity matrix does not change the matrix.

Definition 9. Elementary transformations matrices are called the following operations:

Swap two rows (columns).

Multiply each element of a row (column) by a non-zero number.

Adding to the elements of one row (column) the corresponding elements of another row (column).

Definition 10. The matrix B obtained from the matrix A with the help of elementary transformations is called equivalent(denoted BA).

Example 1.1. Find a linear combination of matrices 2A–3B if

,
.

,
,

.

Example 1.2. Find product of matrices
, if

.

Solution: since the number of columns of the first matrix is ​​the same as the number of rows of the second matrix, then the matrix product exists. As a result, we get a new matrix
, where

As a result, we get
.

Lecture 2. Determinants. Calculation of determinants of the second, third order. Qualifier propertiesn-th order.

Various actions are performed on such matrices: they are multiplied by each other, determinants are found, 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 is named as if it were a single variable. To clarify which of the array cells is meant, when it is mentioned in the program, together with the variable, the cell number in it is used. Both a two-dimensional matrix and an n-dimensional array in the program can contain not only numerical, but also symbolic, string, Boolean and other information, but always the same within the entire array.

Matrices are denoted by 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 - the corresponding lowercase letters with indices indicating their number in the row and in the column a (m, n).

The most common matrices are rectangular, 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 column. 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 zero - 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.

Related videos

Instruction. For online solutions you need to specify a matrix expression. At the second stage, it will be necessary to clarify the dimensions of the matrices.

Valid operations: multiplication (*), addition (+), subtraction (-), matrix inverse A^(-1) , exponentiation (A^2 , B^3), matrix transposition (A^T).
To perform a list of operations, use the semicolon (;) separator. For example, to perform three operations:
a) 3A + 4B
b) AB-BA
c) (A-B) -1
will need to be written like this: 3*A+4*B;A*B-B*A;(A-B)^(-1)

A matrix is ​​a rectangular numerical table with m rows and n columns, so the matrix can be schematically represented as a rectangle.
Zero matrix (null matrix) is called a matrix, all elements of which are equal to zero and denote 0.
identity matrix is called a square matrix of the form

Let's define basic operations on matrices.

Matrix addition

Example 6 . .
The operation of matrix addition extends to the case of any number of terms. Obviously, A+0=A .
We emphasize once again that only matrices of the same size can be added; for matrices of different sizes, the addition operation is not defined.

Matrix subtraction

Matrix multiplication

Example 7 . Matrix data and . Find the matrices C = A·B and D = B·A.
Solution. First of all, note that the product A B exists because the number of columns in A is equal to the number of rows in B.

Example 8 . Given a matrix . Find 3A 2 - 2A.
Solution.

.
; .
.
We note the following curious fact.
As you know, the product of two non-zero numbers is not equal to zero. For matrices, such a circumstance may not take place, that is, the product of nonzero matrices may turn out to be equal to the zero matrix.

The matrix is ​​denoted by capital Latin letters ( BUT, AT, FROM,...).

Definition 1. Rectangular table of the form ,

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 of a matrix BUT 1 is the number denoted by ∆ and equal to , where

Example. Calculate the 2nd order determinant:

Definition 2. Determinant of the 3rd order of a square matrix BUT 2 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 n-th order determinant.

Properties of determinants:

The determinant does not change during transposition (i.e., if the rows and columns in it are interchanged while maintaining the order).

If any two rows or two columns are interchanged in the determinant, then the determinant only changes the sign.

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

If all elements of any row (column) of the 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 does not change if the corresponding elements of another row (column) multiplied by the same number are added to the elements of any row (column).

Example.

Definition 4. The determinant obtained from a given by deleting a column and a row is called minor the corresponding element. M ij element a ij .

Definition 5. Algebraic addition element a ij , is called the expression

§3. Matrix Actions

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 BUT to matrix AT 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 BUT to the corresponding elements of the jth column of the matrix AT. Matrix product BUT to matrix AT can only be found if the number of columns of the matrix BUT equals the number of matrix rows AT. Otherwise, the work is impossible.

Comment:

(not subject to the commutativity property)

§ 4. Inverse matrix

The inverse matrix exists only for a square matrix, and the matrix must be nonsingular.

Definition 1. Matrix BUT called non-degenerate if the determinant of this matrix is ​​not equal to zero

Definition 2. BUT-1 called inverse matrix for a given non-singular square matrix BUT, if when multiplying this matrix by the given one both on the right, then on the left, the identity matrix is ​​obtained.

Algorithm for calculating the inverse matrix

1 way (using algebraic additions)

Example 1:

Note that the elements of the matrix can be not only numbers. Imagine that you are describing the books that are on your bookshelf. Let your shelf be in order and all the books stand in strictly defined places. The table that will contain the description of your library (according to the shelves and the sequence 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 among themselves by some dependence. The resulting table will also be called a matrix. In other words, Matrix is ​​any rectangular table made up of homogeneous elements. Here and below we will talk about matrices composed of numbers.

Instead of parentheses, matrices are written using 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 , something about rectangular.

Depending on the values ​​of m and n, there are some special types matrices:

The most important characteristic square matrix is ​​its determinant or determinant, which is composed of matrix elements and is denoted

Obviously, D E =1 ; .

Definition 3. If a , then the matrix A called non-degenerate or not special.

Definition 4. If a detA = 0 , then the matrix A called degenerate or special.

Definition 5. Two matrices A and B called equal and write A=B if they have the same dimensions and their corresponding elements are equal, i.e..

For example, the 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 dimensions of the matrices are the same, but not all elements in the same places are equal. The matrices are different because they have different sizes. The first matrix is ​​2x3 and the second 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 stand on different places in every matrix. But the matrices and are equal, according to Definition 5.

Definition 6. If we fix a certain number of matrix columns A and the same number of its rows, then the elements at the intersection of the specified columns and rows form square matrix n- th order, the determinant of which called minor k- th order matrix A.

Example. Write out three minors of the second order of the matrix