Pods with matrices. Matrices The rating matrix has the same number of columns
Viznachennya. The size matrix is a table of numbers that is formed by 
rows and 
stovptsiv. The numbers that form the matrix are called matrix elements.
Matrices are designated by the great writers of the Latin alphabet (for example A, B, C), and the matrix elements are represented by small letters with continuous indexing: 
, de 
- row number, 
- Station number.
For example, matrix 
,
or for a short recording 
, de 
;
.
See the matrix.
A matrix that consists of one row is called matrix (vector)-row, and from one hundred – matrix (vector)-stovpc:
- row matrix;
-matrix-stovpets.
The matrix is called square 
-
th order, since the number of rows is equal to the number of columns and equal to 
. For example, 
- A third-order square matrix.
Matrix elements 
, in which the row number is prior to the station number 
, are called diagonal and create head diagonal matrices.
If all non-diagonal elements of a square matrix are equal to zero, then the matrix is called diagonal. For example,
-Diagonal matrix of the third order.
Like a diagonal matrix 
In the second order, all diagonal elements become one, then the matrix is called single matrix 
-th order i won is designated by the letter 
. For example, 
- Alone matrix of the third order.
Operations on matrices.
For example, I guess 
, That 
.
For example: 
,
,
.
butt. Calculate the additional matrix
,
de
;
.
We know the size of the matrix-creation (as a multiplied matrix can be used): 
. The elements of the matrix are computable 
. element 
go out when there is a lot of 
th row of the matrix 
on 
-th column of the matrix 
.
Ottrimomo 
.
,
.
The value is due to the fact that the matrix has a size 
, then the matrix is transposed 
size 
.
For example: 
;
.
Secondary values of square matrices
The primary is a number that characterizes a square matrix.
Leader of the matrix 
signified 
or else 
.
First-order co-sign of the matrix
, or first in line, the element is called 
:
. For example, let's go 
then 
.
Secondary matrix of a different order
, or second-in-command of a different order, is called a number that is calculated using the formula:
.
Create 
і 
are called members of the executive committee another order. For example, let's go 
then 
.
Let us be given a third-order square matrix:
.
Secondary matrix of the third order, or third-order leader, is called a number that is calculated using the formula:
This number is the sum of algebra, which is composed of 6 additions, or 6 terms of the denominator. Skin additions include exactly one element from each skin row and skin column of the matrix. The signs that are included in the formula before the formula are easy to remember by following the diagram (Fig. 1.), which is called rule of tricutaneous or else Sarrus rule.
In order to calculate the principal figures of higher orders, certain additional concepts are required.
Let me give you a square matrix 
n-th order.
Minor 
element 
matrices n The digit of the matrix is called in the th order ( n–
1)th order, extracted from the matrix 
Vikresluvannyam 
-th row 
-th stovptsya.
For example, element minor 
matrices 
third order will be:
Algebraic additions 
element
matrices n The th order is called the th minor, with the sign 
:
, then. The addition of algebra is avoided with a minor if the sum of the row numbers and column ( i+
j) is a number, and is subdivided into a minor sign if ( i+
j) is an odd number. For example, ; 
.
To calculate the primary values of square matrices higher than the third order, Laplace's theorem is used.
Laplace's theorem.The derivative of a square matrix is the same as the sum of the creative elements of any row (row) in their algebra:
(Sorting out the elements i-
th rows; 
);
(Sorting out the elements j-
go stovptsya; 
);
Under the authority of the primary ones, the primary matrix of the matrix cannot be changed unless the elements of any row (row) of the matrix are supplemented with elements of another row (row), first multiplied by the same number. This power of primary figures and Laplace’s theorem allow us to essentially simplify the calculation of primary sources of high orders. When calculating the variables, it is necessary to rearrange the output matrix so that the matrix is rearranged in a small row (or row) to contain the most zeros, and then find the output matrix for the row (or column).
butt. Calculate the fourth-order variable:
.
Let's rearrange the matrix so that all elements of the 3rd row, except one, turn to 0. For which multiply the elements of the 3rd column by (-4) and by 2 and add them to the elements of the 1st and 2nd columns . Unfolding the otmannik behind the elements of the third row, we know
.
The third-order derivation can be calculated using the Trikutnik rule or Laplace's theorem, however, the matrix can be simplified further. We “zero” the third-order matrix elements of the 2nd row (except one). For which element of the third column of the matrix, first multiplied by (-13) and by 4, is added to the elements of the 1st and 2nd columns in a similar way:
.
Arranging on the elements of another row and wine-filled multipliers, we can remove them.
Danish methodical guide to help you learn how to write diies with matrices: added (extracted) matrix, transposed matrix, multiplied matrix, found return matrix. All the material is presented in a simple and accessible form, with similar applications, so that people who are probably unprepared can learn to compose actions with matrices. For self-monitoring and self-checking, you can easily use a matrix calculator >>>.
I will try to minimize theoretical calculations, avoid possible explanations “at the fingertips” and use a variety of non-scientific terms. Lovers of ground theory, kindly, do not engage in criticism, our task is learn how to work with matrices.
For advanced preparation of the topic (who is “on fire”) there is an intensive pdf course The Matrix, the originator and the zalik!
The matrix is a straightforward table of beings elements. In yakosti elements We look at numbers as numerical matrices. ELEMENT- This is the term. The term must be memorized, it will often appear in bold, so I don’t accidentally use bold font to show it.
Designated: matrices are called great Latin writers
Butt: Let's look at the two-by-three matrix:
This matrix consists of six elements:
All numbers (elements) in the middle of the matrix appear on their own, so there is nothing special about them: 
It's just a table of numbers!
Also homey don't rearrange numbers, as otherwise not stated in the explanations. Each number has its own place of reshuffling, and it is not possible to shuffle them!
The matrix has two rows: 
and three points: 
STANDARD: if we talk about the dimensions of the matrix, then from the beginning indicate the number of rows, and then the number of columns. Only a few of us picked out the “two by three” matrix using brushes.
If the number of rows and columns of the matrix is eliminated, then the matrix is called square, for example: 
- a three-by-three matrix.
If there is one column or one row in a matrix, such matrices are also called vectors.
In fact, we already know the concept of a matrix, let’s look at, for example, the point with coordinates “x” and “player”: . In fact, the coordinates of the point are recorded in a one-by-two matrix. Before speaking, the axis and butt of you, which is why the order of numbers has meaning: i – there are two different points of the plane.
Now let’s move straight to the Vivchennya action with matrices:
1) Diya persha. Adding minus from the matrix (introducing minus to the matrix).
Let's return to our matrix 
. As you melodiously noted, this matrix has a lot of negative numbers. It’s very difficult to visualize different actions with the matrix, it’s difficult to write so many minuses, and it just looks ugly when designed.
There is a minus for the boundaries of the matrices, changing the sign of the SKIN element of the matrix:
At zero, as you know, the sign does not change, zero is in and in Africa zero.
Buttstock: 
. Looks condescending.
We introduce a minus sign into the matrix by changing the sign of the SKINAL element of the matrix:
Well, it turned out much nicer. And, most importantly, it will be EASIER to connect any actions with the matrix. Because there is such a mathematical folk saying: the more minuses, the more confusion and compromises.
2) A friend's activity. Multiplying a matrix by a number.
Butt:
It's simple, just multiply the matrix by the number you need skin multiply the matrix element by a number. Whose choice is a three.
Another brown butt:
– multiplication of matrix by fraction
Let's take a look at those who are shy right now NOT REQUIRED:
There is NO REQUIREMENT to enter into the matrix; first of all, it compiles further actions from the matrix; in other words, it complicates the verification of the decision made by the depositor (especially when 
- Residual confirmation of knowledge).
Tim more NOT REQUIRED divide the skin element of the matrix by minus sim:
Zi statti Mathematics for dummies or where to start We remember that tens of fractions are completely different from other mathematics.
One thing bazhano work in this application - add a minus matrix:
And from Yakbi ALL matrix elements were divided by 7 no extra charge, It is possible (and required!) to be divided.
Butt:
In what way can I REQUIRED multiply all matrix elements by , so that all matrix numbers are divided by 2 no extra charge.
Note: theoretically, in high school mathematics, there is no “floor” concept. Instead of the phrase “tse divided by tse,” you can now say “tse multiplied by drib.” This is the hem – this is a huge fall of multiplication.
3) Act three. Matrix transposition.
To transpose a matrix, you need to write its rows in the column of the transposed matrix.
Butt:
Transpose the matrix
There is only one row here and, as a rule, it needs to be written down in the column:
– matrix is transposed.
The transposed matrix is indicated by a superordinate index or a right-handed stroke.
Pokrokovy butt:
Transpose the matrix 
From now on, let’s rewrite the first row of the first column:
Then we rewrite another row in another column: 
I, we found, we rewrite the third row in the third column:
Ready. Roughly speaking, transposition means turning the matrix on its side.
4) Day of the fourth. Suma (sacrifice) matrix.
The sum of the matrix is awkward.
NOT ALL MATRIX CAN BE FOLDED. For the visualization of the folded matrix, it is necessary for the smells to be the SAME OUTSIDE THE WORLD.
For example, if you are given a two-by-two matrix, then you can add only the two-by-two matrix and the other! 
Butt:
Matrix folds 
і 
In order to fold the matrix, it is necessary to fold their supporting elements:
For difference, the matrix rule is similar, it is necessary to know the difference between the different elements.
Butt:
Find out the difference matrix 
, 
What is the simplest way to navigate this butt without getting lost? There will be a lot of minuses, for which we add a minus to the matrix:
Note: theoretically, in high school mathematics, the school concept of “discovery” is absent. Instead of the phrase "from whom to take this" you can always say "to whom add a negative number." This is a very difficult thing to do.
5) Day five. Reproduction matrix.
What matrices can be multiplied?
So the matrix can be multiplied by the required matrix, so that the number of columns in the matrix is equal to the number of rows in the matrix.
Butt:
How can you multiply a matrix by a matrix?
Then, you can multiply the matrix data.
And if the matrices are rearranged, then in this case the multiplication is already impossible!
Ozhe, visconati multiplied unbearably:
It is no longer so rare to encounter tricky problems when a student is forced to multiply matrices, the multiplication of which is obviously impossible.
It should be noted that for a number of cases it is possible to multiply matrices this way and that way.
For example, for a matrix, it is possible to multiply as well as multiply
Value 1. Matrix A sizem n is called a rectangular table with m rows and n columns, which consists of numbers or other mathematical expressions (called matrix elements), i = 1,2,3, ..., m, j = 1,2,3, ..., n.
, or
Value 2.
Two matrices 
і 
are called the same size equal, if they are solved element by element, then. 
=
, i = 1,2,3, ..., m, j = 1,2,3, ..., n.
Using an additional matrix, it is easy to write down economic activities, for example, tables of the division of resources according to various areas of the economy.
Value 3. If the number of rows of the matrix is reduced from these columns, then. m = n, then the matrix is called square ordern, and in another case straight-cut.
Value 4. The transition from matrix A to matrix A t, in which rows and columns have been swapped out of order, is called transpose baths matrices.
See the matrix: square (size 33) - 
,
straight-cut (size 25) - 
,
diagonal - 
, single - 
, nullova - 
,
matrix-row - 
, matrix-stovpets -.
Vicenance 5.
Elements of a square matrix of order n with the same indices are called elements of the head diagonal, then. tse elementi: 
.
Vicenance 6. Elements of a square matrix of order n are called elements of the side diagonal, since their indices are equal to n + 1, then. ce elementi: .
1.2. Operations on matrices.
1
0
.
Sumoyu
two matrix 
і 
However, the matrix C = (z ij) is called the same size, the elements of which are equal to ij = a ij + b ij (i = 1,2,3,…,m, j = 1,2,3,…,n).
The power of the operation is a folded matrix.
For any matrix A, B, Z of the same size, equalities are determined:
1) A + B = B + A (comutativity),
2) (A + B) + C = A + (B + C) = A + B + C (associativity).
2
0
.
Creator
matrices
per number
called matrix 
the same size as matrix A, and b ij = 
(i = 1,2,3, ..., m, j = 1,2,3, ..., n).
The power of the operation of multiplying a matrix by a number.
(A) = ()A (associativity of multiplication);
(A+B) = A+B (distributivity of multiplication before folding matrix);
(+)A = A+A (distributivity of multiplication and addition of numbers).
Vicenza 7.
Linear combination matrix 
і 
However, the new size is called the type A+B, where and are complementary numbers.
3 0 . Dobutcom A U matrix And I vidpovly Rosemirv mn і nk edit matrix ryzzira mk, taka, pusel Element Z ij Dorivnu, Sumy Creative Elementiv I-Togo Row Avo, Jo Stovptsia Matrix V, Tobto. z ij = a i 1 b 1 j + a i 2 b 2 j + ... + a ik b kj .
Addition AB works only because the number of columns of matrix A is equal to the number of rows of the matrix.
The power of the operation of the multiplied matrix:
(AB)C = A(BC) (associativity);
(A+B)C = AC+BC (distributivity of the folded matrix);
A(B+C) = AB+AC (distributivity of the folded matrix);
ABBA (not commutative).
Vicenza 8. Matrices A and B, for which AB = BA, are called commuting or commuting.
Multiplying a square matrix of any order by a unique singular matrix does not change the matrix.
Vicenza 9. By elementary re-creations the matrix is called the following operations:
Swap two rows (stovpts).
Multiplying the skin element of the row (stovptsya) by a number subtracted from zero.
Adding to elements of one row (stack) similar elements of another row (stack).
Viznachennya 10. The matrix, taken from the matrix And with the help of elementary transformations, is called equivalent(indicated by BA).
butt 1.1. Find the linear combination of matrix 2A-3B, which
,
.
,
,
.
butt
1.2.
Find out the additional matrix 
, yakscho
.
Solution: if a number of rows of the first matrix are combined with a number of rows of another matrix, then the matrix is created. As a result, a new matrix is created 
, de
As a result, we reject 
.
Lecture 2. Secondaries. The calculation of the derivations is of a different, third order. The power of the leadersn-th order.
Matrix. Pods over matrices. The power of operations on matrices. See the matrix.
Matrices (and a particularly mathematical branch – matrix algebra) There are more important implications for applied mathematics, which allows us to write down in a simple form a significant part of mathematical models of objects and processes. The term "matrix" appeared in 1850. Matrices were first figured out in ancient China, and later in Arab mathematicians.
Matrix A=A mn order m*n called There is a straightforward table of numbers that shows m - rows and n - rows.
Matrix elements aij, in which i=j are called diagonal and equal head diagonal.
For a square matrix (m=n), the head diagonal is formed by the elements a 11, a 22,..., a nn.
Jealousy matrix.
A=B what is the order matrix Aі B however a ij = b ij (i=1,2,...,m; j=1,2,...,n)
Pods over matrices.
1. Added matrix - element-by-element operation
2. Removable matrix - element-by-element operation
3. Adding a matrix to a number - an elemental operation
4. Polygamy A*B matrix by rule row on Stovpets(the number of rows of matrix A may be equal to the number of rows of matrix B)
A mk * B kn = C mn What does the skin element have to do with it? з ij matrices Cmn the corresponding sums of the creation of elements of the i-th row of matrix A and the corresponding elements of the j-th row of matrix B, then.
Let's demonstrate the matrix multiplication operation in practice
5. Zvedenya at the steps
m>1 is an integer positive number. A is a square matrix (m=n). especially relevant for square matrices
6. Transposition of matrix A. Transpose of matrix is designated A T or A
The rows and stations changed places
butt
The power of operations on matrices
(A+B)+C=A+(B+C)
λ(A+B)=λA+λB
A(B+C)=AB+AC
(A+B)C=AC+BC
λ(AB)=(λA)B=A(λB)
A(BC)=(AB)C
(λA)"=λ(A)"
(A+B)"=A"+B"
(AB)"=B"A"
See the matrix
1. Straight cut: mі n- more positive numbers
2. Square: m=n
3. Matrix row: m=1. For example, (1 3 5 7) - in many practical applications such a matrix is called a vector
4. Stovpets matrix: n=1. For example
5. Diagonal matrix: m=nі a ij = 0, yakscho i≠j. For example
6. One by one matrix: m=nі
7. Zero matrix: a ij =0, i=1,2,...,m
j=1,2,...,n
8. Tricutaneous matrix: all elements below the head diagonal reach 0.
9. Symmetric matrix: m=nі a ij = a ji(so that equal elements stand on symmetrical places along the head diagonal), and then A"=A
For example,
10. Skew-symmetric matrix: m=nі a ij =-a ji(Then the prostrate elements should be placed at symmetrical positions along the head diagonal). Well, on the head diagonal there are zeros (but when i=j maєmo a ii =-a ii)
I realized A"=-A
11. Hermite matrix: m=nі a ii =-ã ii (ã ji- comprehensively - received before a ji, then. yakscho A=3+2i, then comprehensively - obtained Ã=3-2i)
