MATRIX I card holder

Lecture 1. Matrix

1. Understanding the matrix. tipi matrix

2. Matrix algebra

Lecture 2. Business cards

1. Designers of a square matrix and power

2. Laplace and annuity theorems

Lecture 3. Zvorotn_y matrix

1. Understand wrapped matrix... Uniqueness of the wrapped matrix

2. Algorithm to induce the wrapped matrix. The power of the wrapped matrix

4. Zavdannya i right

4.1. Matrix and diy above them

4.2. Viznachniki

4.3. ring matrix

5.Individual staff

literature

Lecture 1. MATRIX

plan

1. Understanding the matrix. Tipi matrices.

2. Algebra of matrices.

The Keys of the Intelligencer

Diagonal matrix.

Single matrix.

Zero matrix.

The matrix is ​​symmetric.

Usgodzhenist matrix.

Transponding.

Tricut matrix.

1. UNDERSTANDING THE MATRIX. TYPE matrix

straight table

to be stored in m rows and n hundredpoints, elements of each and every number, de i- row number, j- the number of the hundredths on the overturns which are worth the element, we will name the numerical matrix order m'n and mean.

The main types of matrices are visible:

1. Let m = n, todі matrix A - square matrix, yaka maє order n:

A = .

elements make a headache diagonal, elements I will make a bit of a diagonal.

diagonal , Yaksho all the elements, crim, you can, the elements of the head diagonal, get to zero:

A = = diag ( ).

Diagonal, and hence square, the matrix is ​​called lonely , For all the elements of the head diagonal equal to 1:

E = = diag (1, 1, 1, ..., 1).

It’s amazing that a single matrix is ​​a matrix analogue of one in unlawful numbers, and it is also acceptable that a single matrix is ​​used only for square matrices.

Put single matrices at a glance:

Square matrices

A = , B =

are called upper and lower tricky ones.

2 ... Come on m = 1, todi matrix A is a row matrix, yaka maє viglyad:

3 ... Nokhai n = 1, todi matrix A is a matrix-hundred-point, yaka maє viglyad:

4 A zero matrix is ​​a matrix of order m'n, all elements of which are equal to 0:

Fascinatingly, a zero matrix can be square, a row matrix or a stove matrix. Zero matrix є matrix analogue of zero in arbitrary numbers.

5 ... matrix be called transposed before the matrix and it is known as the number of rows of the matrix.

butt . Nekhai =, todi =.

Remarkably, if the matrix A is of order m'n, then the transposed matrix is ​​of order n'm.

6 ... Matrix A be called symmetric , Yaksho A = A, i oblique , Yaksho A = -A.

butt . Follow-up on the symmetry of the A and B.

Todi =, also, the matrix A is symmetric, so yak A = A.

B =, todi =, the same, matrix B is skew-immetric, so yak B = - B.

Impressively, it is symmetrical and skewed matrix depending on the square. On the head diagonal of a symmetric matrix, one can stand whether it is elements, and symmetrically, the head diagonal is guilty of standing the same elements, tobto =. On the head diagonal of the skewed matrix, it should be zero, and symmetrically, the head diagonal = -.

2. ALGEBRA matrix

It is easy to see the action over the matrices, but a few words can be introduced to understand the new ones.

Two matrices A and B are called matrices of the same order, as they smell the same number of rows and the same number of hundred.

Butt. i - matrices of the same order 2'3;

I - matrices of different orders, so yak 2'3 ≠ 3'2.

Understanding "more" and "less" for matrices does not exist.

Matrices А and В are called equal, as they smell of the same order m'n, і =, de 1, 2, 3, ..., m, and j = 1, 2, 3, ..., n.

Matrix multiplication by number.

Multiply the matrix А by the number λ to multiply the skin element of the matrix by the number λ:

λА = , ΛR.

Given the value of the matrix, the zagalny multiplier of all elements in the matrix can be blamed for the sign of the matrix.

Butt.

Come on matrix A =, todi 5A = =.

Come on matrix B = = = 5.

The power of multiplying a matrix by a number :

2) (λμ) А = λ (μА) = μ (λА), de λ, μ R;

3) (λА) = λА;

Sum (business) matrix .

Sum (difference) is only used for matrices of the same order m'n.

The sum (difference) of two matrices A and in order m'n is a matrix of the same order, de = ± (1, 2, 3, ..., m ,

j= 1, 2, 3, ..., n.).

In other words, matrix C is stored in elements, sums (differences) of similar elements in matrices A and B.

butt . Know the sum and the difference of the matrix A and B.

todі = + = =,

=–==.

Yaksho f = , =, To A ± B is not like a matrix of different order.

Z danih vische viznachen follow power sumi matrix:

1) commutability A + B = B + A;

2) association (A + B) + C = A + (B + C);

3) distributivity up to multiplication by the number λR: λ (А + В) = λА + λВ;

4) 0 + A = A, de 0 is a zero matrix;

5) A + (- A) = 0, de (-A) - matrix, opposite to matrix А;

6) (A + B) = A + B.

Tvir matrices.

The operation “create” is not meant for all matrices, but rather for the narrow ones.

Matrix А and В are called uzgojenimi , If the number of 100 matrices A is equal to the number of rows in the matrix B. So, if ,, m ≠ k, then matrices A and B uzgozheni, so n = n, and in the rotary order of matrices B and A unsuitable, so m ≠ k. Square matrix matrices, if they have the same order n, moreover, they have the same order as A and B, as well as B and A. If, then, there will be used matrices A and B, and also matrices B and A, since n = n, m = m.

Two uzgodzhenih matrices curd і

A = , B =

be called a matrix 3 of order m'k:

= ∙, elements which are calculated according to the formula:

(1, 2, 3, ..., m, j = 1, 2, 3, ..., k),

that is the element of the i -th rows and the j-th hundredth of the matrix From the back sum of the creations of all the elements of the i -th rows of the matrix A on the specific elements of the j-th hundred of the matrix B.

butt . Know the tvir matrices A and B.

∙===.

Tvir matrix В ∙ А is not ісує, so the matrix В і А not uzgodzheni: matrix В is order 2'2, and matrix А - order 3'2.

Clear power create a matrix:

1 ) Non-commutative: AB ≠ VA, navit like A і B, і B і A uzgozhenі. If AB = BA, then the matrices A and B are called commuting (matrices A and B are usually square).

butt 1 . = , = ;

==;

==.

Obviously,.

butt 2 . = , = ;

= = =;

= = = .

visnovok: ≠, if the matrices i are of the same order.

2 ) For any square matrices, a single matrix E commuting to a matrix A of the same order, and in the result we can deduce the same matrix A, so that AE = EA = A.

butt .

===;

===.

3 ) A 0 = 0 A = 0.

4 ) Twir two matrices can be zero, for the same matrix A and B they can be non-zero.

butt .

= ==.

5 ) Association ABC = A (BC) = (AB) C:

· (·

butt .

Mamo Matrix, , ;

todi A ּ (B ּ C) = (

(A ּ B) ּ C =

===

==.

In such a rank, mi on the butt showed that A ּ (B ּ C) = (A ּ B) ּ C.

6 ) Distributive nature of folding:

(A + B) ∙ C = AC + BC, A ∙ (B + C) = AB + AC.

7) (A ∙ B) = B ∙ A.

Butt.

, =.

Todi AB =∙==

=(A ∙ B)= =

V ∙A =∙ = ==.

In such a rank, ( A ∙ B)= V A .

8 ) Λ (A ּ B) = (λA) ּ Y = A ּ (λB), λ, R.

Just put on the display over the matrices, so that you need to know the bag, the price, the two matrices A and B.

butt 1 .

, .

Decision.

1) + = = =;

2) – ===;

3) Tvir is not іnu, so like a matrix A and In unsuitable, however, not іnu і create for this reason.

butt 2 .

Decision.

1) sum the matrices, like і іх differences, not існіє, so as the matrixes of different order: matrix A is of order 2'3, and matrix B is order 3'1;

2) so as matrices A and B uzgodzheni, then tvir matrices A ּ U isnu:

·=·= =,

Tvir matrix B ּ A not ісу, so as matrices and ineffectiveness.

Butt 3.

Decision.

1) sum the matrices, like і іх differences, not існіє, so as the matrixes of different order: matrix A is order 3'2, and matrix B - order 2'3;

2) a set of matrices A ּ Y, so і B ּ A, ісу, so as matrices і uzgodzhenі, or the result of such creations will be matrices of different orders: · =, · =.

= = ;

·=·= =

In this vypadku AB ≠ VA.

butt 4 .

Decision.

1) +===,

2) –= ==;

3) tvir yak matrix A ּ V, so i V ּ A, Isnu, so like the matrix of uzgodzheni:

·==·= =;

·==·= =

= ≠, so that matrices A and B are noncommuting.

butt 5 .

Decision.

1) +===,

2) –===;

3) twir yak matrices А У, so і В ּ А, ісує, so yak matrices і uzgodzhenі:

·==·= =;

·==·= =

A ּ Y = B ּ A, i.e., these matrices are commuting.

Lecture 2.Viznachnik

plan

1. Designers of a square matrix and power.

2. Laplace's theorem and annuity.

The Keys of the Intelligencer

Algebraic additional elements of the design.

Мінор element of viznachnik.

The business card holder is of a different order.

Business card holder of the third order.

The business card holder is in order.

Laplace's theorem.

Anuluvannya theorem.

1.share square matrix TA ЇX POWER

Nekhai A is a square matrix of order n:

A = .

The skin of such a matrix can be set in the form of a single number, which is called the matrix designer (determinant) and is known

Det A = Δ = .

Apparently, the designer is the only one for square matrix.

The rules for calculating the designators and powers for square matrices of a different and third order are understandable, as we will name the designators of another and third order for the designation.

Business manager of a different order matrices are called a number, like they start after the rule:

that is, the designator of a different order is a number, like a complement of elements of the head diagonal minus tvir of elements of the secondary diagonal.

butt .

Todi == 43 - (-1) 2 = 12 + 2 = 14.

Slide memory, for the matrices of the matrices, the round or square bows are victorious, and for the designer - vertical lines... The matrix is ​​a table of numbers, and the matrix is ​​a number.

For a different order, follow the same order power :

1. The business card holder does not change when changing all of the following rows:

2. The sign of the visitor should be changed to the opposite one when rearranging the rows (100%) of the visor:

3. The initial multiplier of all the elements of the row (one hundred percent) of the visitor can be blamed for the sign of the visitor:

4. If all the elements of the row (100%) of the business card holder are back to zero, then the business card holder is back to zero.

5. Business card holder for zero, as well as for certain elements of the row (one hundred percent) proportions:

6. If there are elements of one row (100%) of a card holder for a sum of two documents, then such a card for a card for a sum of two documents:

=+, =+.

7. The value of the card holder does not change, even before the elements of the first row (hundred) to add (take) the items of the first row (one hundred), multiplied by one or the same number:

=+=,

so yak = 0 by power 5.

The decision of the power of the viznachnik is clearly lower.

Introduce the third-order visitor's assistant: the third order a square matrix is ​​a number

Δ == det A = =

=++– – – ,

that is, the dermal supplement in formula (2) is a set of elements of the formulas, taken one at a time or only one from the skin row and the skin column. To remember, as you create in formula (2) brothers with a plus sign, and if you do with a minus sign, the rule of trikutniks (Sarrus rule):

butt . enumerate the visitor

==

If it means that the authority of the visitor of a different order, the visce is discerned, without any changes, is transferred to the name of the visitor in any order, including the third.

2. Laplace's I ANULYUVANN'S theorem

There are two even more important powers that be seen.

Introduced an understanding minor and algebraic supplement.

Minor element viznachnik to be called a viznachnik, otrimaniy from a vykhid viznachnik vikreslyuvannya of that row and that hundred, which are due to the element. Poznachayut minor element through.

butt . = .

Todi, for example, =, =.

Algebraic additions to the element The visitor is called the yogi minor, the captures are a sign. Algebraic additions will be understood, that is =.

for example:

= , === –,

Let's turn to the formula (2). Grouping elements and wines by the arches are a multiplier, we can recognize:

=(– ) +( – ) +(–)=

Equivalence should be similarly adjusted:

1, 2, 3; (3)

Formulas (3) are called distribution formulas Visitor for the elements of the i-th row (j-th hundred), or Laplace formulas for the visitor of the third order.

In such a rank, we will mo eighth authority :

Laplace's theorem ... Business card holder for the sum of all creations of elements in any row (hundred) on news algebraic supplements elements of a row (100).

Remarkably, the power of the business card holder is given є not the same, as the business card holder is in some order. On the practice of yogo vikoristovuyut for the calculation of the visitor be-in order. As a rule, the first one to count the visitor, the vicarious authorities 1 to 7, it is suggested that, as a rule, in any row (one hundred percent) they added zero all the elements, except for one, and then put them on the elements.

butt . enumerate the visitor

== (From another row I see it) =

== (From the third row I see it) =

== (We will put the form for the elements of the third

rows) = 1 ּ = (from the other tie, see the first hundred) = = 1998 ּ 0 - 1 ּ 2 = -2.

butt .

The fourth-order visor is visible. For the calculation of the speed by the Laplace theorem, in order to spread out the elements of a row (store).

== (so as another hundredth to avenge three zero elements, then it is possible to put the designator behind the elements of another hundredth) = = 3 ּ = (from another row I’ll see it, multiply by 3, and from the third row I’ll see it by 2, = multiply

3 ּ = (We can put the blank behind the elements of the first stack) = 3 ּ 1 ּ =

no power defining the name Anuluvannya theorem :

the sum of all creations of the elements of one row (hundred) of the designer for all algebraic additional elements of the first row (one hundred) is added to zero, so that

++ = 0,

butt .

= = (Placed behind the elements of the third row) =

0 ּ +0 ּ + ּ = -2.

Ale, for a good butt: 0 ּ +0 ּ +1 ּ =

0 ּ +0 ּ +1 ּ = 0.

Yakshho viglyad, be it in order

=That is, it’s worthwhile to get some elements to stand on the diagonal:

= ּּ ... ּ. (4)

Butt. Count the badge.

=

In some cases, when a visitor is counted for the help of an elementary revision, it is given to bring it to a tricted one, for which the formula (4) becomes stagnant.

If you want to create two square matrices, then you need to add a designer to the square matrices :.

Lecture 3. ROTARY MATRIX

plan

1. Understanding the wrapped matrix. Uniqueness of the wrapped matrix.

2. Algorithm to induce the wrapped matrix.

The power of the wrapped matrix.

The Keys of the Intelligencer

Zvorotna matrix.

Matrix is ​​accepted.

1. UNDERSTAND THE ROTARY MATRIX.

ONE SPOTLING MATRIX

In the theory of numbers, the order of the number is the number, which is more opposite to that () also, scho, і number, the same, scho. For example, for the number 5, the opposite will be the number

(- 5), and it will be a number. Similarly, in the theory of matrices, they also introduced the understanding of the prototype matrix, the meaning (- A). ringing matrix for a square matrix A, the order n is called a matrix,

de E is a single matrix of order n.

Immediately and obviously, a vocal matrix is ​​simple only for square non-virtual matrices.

The square matrix is ​​called non-virgin (Nonsingular), if detA ≠ 0. If detA = 0, then the matrix A is called virginia (Special).

Apparently, a non-virtual matrix A has a single rotating matrix. Brought to the tverdzhennya.

Come on for the matrix Aіsnu two anti-matrix matrices, tobto

Todi = ּ = ּ () =

It’s necessary to bring it up.

We know the wrapper of the wrapped matrix. So, as a formatter, add two matrices A and in the same order, add a formatter for two matrices, i.e.

Robimo is a visnovok, which is the form of the wrapped matrix є the number, which is the same as the determinant of the visual matrix.

2. ALGORITHM WAKE UP THE ROTARY MATRIX.

The power of the GREAT MATRIX

It will be shown that if the matrix A is non-virtual, then for her there is a ringing matrix, and I will stay.

It is foldable by a matrix with algebraic additional elements of matrix A:

Transposing її, we recognize so called conferred matrix:

.

We know tvir ּ. With respect to Laplace's theorems and annulment theorems:

ּ = =

=.

Robimo visnovok:

Algorithm to induce the wrapped matrix.

1) Calculate the matrix format A... If the formatting tool is written to zero, then the ringing matrix is ​​not.

2) If the matrix matrix is ​​not suitable for zero, then the addition of algebraic additional elements of the matrix A matrix.

3) By transposing the matrix, remove the assigned matrix.

4) Behind the formula (2), the layers are wrapped with a matrix.

5) For the formula (1) to reconsider the calculation.

butt ... Know the wrapped matrix.

a). Nekhai A =. So, as the matrix A has two identical rows, then the matrix of the matrix is ​​left to zero. Again, the matrix is ​​virogen, and for her it is not a wrapped matrix.

b). hey A =.

Numeric matrix matrix

ringing matrix isnuє.

A foldable matrix with algebraic extras

= = ;

by transposing the matrix, we can recognize the assigned matrix

the formula (2) is known to be inverted by the matrix

==.

The correctness of the calculation is revised

= = .

Again, the ringing matrix is ​​prompted to be correct.

The power of the wrapped matrix

1. ;

2. ;

3. .

4. PRESENTATION І RIGHT

4.1 Matrices and diy above them

1. Know the bag, the retail, create two matrices A and B.

a) , ;

b) , ;

v) , ;

G) , ;

e) , ;

e) , ;

g) , ;

h), ;

і) , .

2. Bring the commuting matrices A and B.

a),; b) , .

3. Given matrix A. B and C. Show that (AB) · C = A · (BC).

a) , , ;

b) , , .

4. Count (3A - 2B) · С, yaksho

, , .

5. Know what

a) ; b) .

6. Know the matrix X, where 3A + 2 X = B, de

, .

7. Know ABC, yaksho

a) , , ;

b) , , .

VIDPOVIDI ON THEMI "MATRIX І DIV OVER THEM"

1.a) , ;

b) create AB and VA not іsnu;

v) , ;

G) , ;

e) sumi, difference and add-on VA matrix is ​​not visible, ;

e), ;

g) do not create matrices;

h) , ;

і) , .

2.a) ; b) .

3.a) ; b).

4. .

5.a) ; b) .

6. .

7.a) ; b) .

4.2 Business cards

1. Count the badges

a); b); v); G); e); e);

g); h) .

3. For the help of the rules of trikutniks, enumerate the visnatniki

a); b); v) ; G).

4. Count the markers for the butt 2, Laplace's vikory theorem.

5. Numerate the viznachniki, after forgiving їх:

a) ; b) ; v) ;

G); e) ; e) ;

g) .

6. Calculate the visnatnik by the method of bringing the yogo to the tricut viglyad

.

7. Let it be given to the matrix A and B. :

, .

VIDPOVIDI ON THEMI "viznachnik"

1.a) 10; b) 1; c) 25; d) 16; e) 0; f) -3; g) -6; h) 1.

2. a) -25; b) 168; at 21; d) 12.

3. a) -25; b) 168; at 21; d) 12.

4. a) 2; b) 0; c) 0; d) 70; e) 18; f) -66; g) -36.

4.3 Zvorotn_y matrix

1. Know the wrapped matrix:

a); b); v); G);

e) ; e); g) ; h) ;

і) ; To) ; l) ;

m) ; n) .

2. Know the wrapped matrix and reconsider the display:

a); b) .

3. Bring the parity :

a),; b) ,.

4. Bring the parity :

a); b) .

VIDPOVIDI ON THEMI "Zvorotniy MATRIX"

1.a); b); v) ; G) ;

e) ; e) ; g);

h) ; і) ;

To) ; l) ;

m); n) .

2.a) ; b) .

2.a) , , =;

b) , ,

=.

5.a) , ,

, ;

b) , ,

, .

5. INDIVIDUAL ZAVDANNYA

1. List the card holder for the distribution

a) in the i-th row;

b) on the j-th column.

1.1. ; 1.2. ; 1.3. ;

i = 2, j = 3.i = 4, j = 1.i = 3, j = 2.

1.4. ; 1.5. ; 1.6. ;

i = 3, j = 3.i = 1, j = 4.i = 2, j = 2.

1.7. ; 1.8. ; 1.9. ;

i = 4, j = 4.i = 2, j = 2.i = 3, j = 2.

1.10. ; 1.11. ; 1.12. ;

i = 2, j = 1.i = 1, j = 2.i = 3, j = 2.

1.13. ; 1.14. ; 1.15. ;

i = 2, j = 3.i = 1, j = 3.i = 4, j = 2.

1.16. ; 1.17. ; 1.18. ;

i = 2, j = 3.i = 2, j = 4.i = 1, j = 3.

1.19. ; 1.20. ; 1.21. ;

i = 2, j = 2.i = 1, j = 4.i = 3, j = 2.

1.22. ; 1.23. ; 1.24. ;

i = 1, j = 3.i = 2, j = 1.i = 3, j = 4.

1.25. ; 1.26. ; 1.27. ;

i = 4, j = 3.i = 3, j = 3.i = 1, j = 2.

1.28. ; 1.29. ; 1.30. .

i = 3, j = 3.i = 2, j = 1.i = 3, j = 2.

LITERATURE

1. Zhevnyak R.M., Karpuk A.A. Vishcha mathematician. - Pl.: Compulsory. shk., 1992.- 384 p.

2. Gusak A.A. Dovidkovy prospect to the revision of the plant: analytical geometriya and line algebra. - Minsk: TetraSystems, 1998. - 288 p.

3. Markov L. N., Razmislovich G. P. Vishcha mathematician. Chastina 1. Minsk: Amalfeya, 1999. - 208 p.

4. Belke IV, Kuzmich K.K. Vishcha mathematics for economics. I semester. M .: Nové znannya, 2002. - 140 p.

5.Kovalenko N.S., Minchenkov Yu.V., Ovseets M.I. Vishcha mathematician. Textbook. posibnik. Minsk: CHIUP, 2003 .-- 32 p.

The main numerical characteristic of a square matrix is ​​a design form. A square matrix of a different order can be seen

A viznachnik or a determinant of a different order is a number calculated by the offensive rule

by the way,

The now visible third-order square matrix

.

The third-order viznachnik is the number calculated by the offensive rule

I will mark the memory of the day before, enter at the virazi for a third-order visitor, call the vicorist Sarrus rule: The first three steps are to enter into the right part with a plus sign є two elements, one must stand on the head diagonal matrix, and the skin of the two is a pair of elements, but lie on a parallel line to the bottom of the matrix.

The remaining three warehouses, which enter with a minus sign, start with an analogous rank, only a little bit diagonal.

butt:

The main powers of the matrix

1. The size of the matrix does not change when the matrix is ​​transposed.

2. When rearranging the rows of rows in or out of the matrix, the matrix will change the sign, and the absolute value will be saved.

3. A business card holder, to avenge the proportional rows, or even to zero.

4. Zagalny multiplier of elements of the same row;

5. If all the elements of a certain row, for example, must be back to zero, then the business card itself is back to zero.

6. Even before the elements of an adjacent row, because the size of the designator does not change.

minor matrices are called a template, which refers to a square matrix of the same number of hundreds and rows.

If all the minor order is higher, as it can be folded into the matrix, it will be zero, and the middle of the minor order would be one type of zero, then the number will be called rank the matrix of the matrix.

algebraic additions We will name the element in the order, we will nominate this minor in order, possessions of the same row and hundred, on the re-set, varto element, the captures with a plus sign, as the sum of the indices is familiar to each other.

This rank

,

de first order.

Calculation of the matrix form holder by way of distribution for the elements of a row abstain

Matrix brochure for additional items of any row (whether a stack) of a matrix on a variety of algebraic additional elements of a row (stack). When calculating the matrix designator in this way, follow the following rule: select a row with the greatest number of zero elements. Tsey priyom is allowed to speed up the calculation.

butt: .

When numbered viznachniki, skoristalis by taking the yogi for the elements of the first store. It can be seen from the induced formula, there is no need to calculate the remainder from the formulas in a different order, so that it must be multiplied by zero.

Enumeration of the wrapped matrix

In case of viral matrix ravnyans, it is widespread to form a vortex matrix. Here is a singing world to replace the operation of the day, as in the explicit view in the algebra of matrices in the day.

Square matrices of the same order, which are even single matrix, are called mutually inverse or reversed. The meaning of the rotary matrix is ​​for it is fair

It is possible to enumerate a ring-shaped matrix only for such a matrix, for such a matrix.

Classical algorithm for calculating the ring matrix

1. Write the matrix transposed to matrix.

2. We will replace the skin element of the matrix with a designer, we will deny it as a result of the addition of a row and 100%, on the repetition of the replacement of the element.

3. Tsei viznachnik supervodzhuyut with a plus sign, like the sum of indexes of the element of a pair, and with a minus sign - in the first vipad.

4. Dilate the rendered matrix onto the matrix format.

Most of the mathematical models in the economy are described for an additional matrix and matrix number.

matrix - a straight-line table, like revealing numbers, functions, equal to one of the mathematical objects, roztashovany in rows and hundreds.

Ob'єkti, how to become a matrix, call it її elements ... Matrix is ​​known by the great Latin letters

and їх elementi - malimi.

symbol
means, шо matrix maє
row i stovptsiv, the element, which is located on the peretina first row i -th hundred
.

.

Seems like a matrix A door matrices V : A = B, As it smells like the same structure (that is, the same number of rows and 100%)
, for all
.

Private view matrices

In practice, matrices of a special kind are often seen. Deyakі methods also transfer matrices from one type to another. Most often, they are shown and matrices are hovered below.

	square matrix, number of rows n for the number of hundred n
	matrix-stoovpets
	row matrix
	lower triangular matrix
	upper triangular matrix
	null matrix
	diagonal matrix
E =	single matrix E(Square)
	unitary matrix
	step matrix
	empty matrix

Matrix elements, with relative numbers of rows and stovpts, tobto a ii Set up the head diagonal of the matrix.

Operations over matrices.

.

Power of operations on matrices

Specific powers of operations

Yaksho tvir matrix
- isnu, then tvir
maybe not. Looks like,
... So the multiplication of a matrix is ​​not commutative. Yaksho
, then і are called commutative. For example, diagonal matrices are of the same order of commutative.

yaksho
, Then it is not necessary
abo
... So, twir of nonzero matrices can be a zero matrix. by the way

Operation in the steps only for square matrices. yaksho
, then

.

For viznachennyam vvazhayut
, It doesn't matter to show
,
... Apparently,
not slid, scho
.

Polementne razorudzhennya to the level A. m =
.

transposition operation matrices of fields in lieu of rows in matrices in the same way:

,

by the way

,
.

Power of transposition:

Business card holders of that power.

For square matrices, it is often the case of an understanding visitor - numbers that can be counted behind the elements of the matrix from the rules strictly according to the values ​​of the rules. The whole number є is an important characteristic of the matrix and is denoted by symbols

.

matrix template
є її element .

matrix template
to be calculated according to the rule:

so that, with the creation of elements of the head diagonal, there is a set of elements of the pre-existing diagonal.

For calculating viznachnikі in a higher order (
) It is necessary to introduce an understanding of the minor and the algebraic addition of the element.

minor
element name the formatting tool that can be drawn from the matrix , viscous -y row i -th hundredpets.

Matrix visible rozmir
:

,

Todi, for example,

algebraic additions element call yogo minor, multiplication by
.

,

Laplace's theorem: Square matrix card holder for additional items of any row (one hundred) on the algebraic add-ons.

Butt, foldable
for the elements of the first row, we can make:

Remaining the theorem gives a universal way of calculating the formulas in any order, by fixing them in another. Each row (100%) is selected by the one with the largest number of zeros. For example, it is necessary to calculate the fourth order form

In this vypadku you can place the card in the first place:

For the rest of the row:

The whole butt will show the same way as the upper triangular matrix for additional diagonal elements. It is not important to bring, that the pattern is fair for any three-piece and diagonal matrices.

Laplace's theorem gives the possibility of calculating the value -th order before calculation viznachnikiv
-th order і, in the end bag, before the registration of the visa holder in a different order.

square matrix A order n you can insert the number det A(Abo | A|, Abo), called її visitor , let's step up the rank:

matrix template A also called її determinant ... The rule for calculating the determinant for the matrix order Nє Fold-up foldable for sleeping and sucking. However, there are methods that allow the implementation of the calculation of the design in higher orders on the basis of the design in the lower order. One of the methods of filing on the power of distributing the visitor for the elements of the deyakiy series (power 7). At the same time, it is very respectable that the markers of non-high orders (1, 2, 3) are well-numbered to be counted according to the values.

The calculation of the 2nd order form is illustrated by the scheme:

Application 4.1. Know the matrix labels

When the third-order visitor is numbered, manually register rule of trikutnikiv (Abo Sarrus), as symbolically it can be written like this:

Application 4.2. Calculate the matrix format

det A = 5*1*(-3) + (-2)*(-4)*6 + 3*0*1 — 6*1*1 — 3*(-2)*(-3) — 0*(-4)*5 = -15+48-6-18 = 48-39 = 9.

I will formulate the basic power of the businessmen, the power of the determinants of all orders. Deyakі z tsikh of authority can be explained on the form of the third order.

power 1 ("Equal parity of rows and hundreds"). The business card holder does not change, if the rows are replaced by stovpchiks, and navpaki. In other words,

Nadal rows and hundred will be just nazivat rows of business card holder .

power 2 ... When rearranging two parallel rows in the form, the sign changes.

power 3 ... Viznachnik, which is two identical in a row, dorіvnyuє to zero.

power 4 ... The zagalny multiplier of elements of any number of the visitor can be blamed for the sign of the visitor.

3 powers 3 and 4 next, If all the elements of a series are proportional to those elements parallel to the series, then such a design template is zero.

really,

power 5 ... As well as elements of whether there are a number of business cases є sums of two donations, then a business card can be placed on a bag of two types of business cards.

by the way,

Power 6. ("An elementary re-creation of the designer"). The business card holder does not change, as far as the elements in one row are given, the same elements are parallel to the row, multiplied by the number.

butt 4.3... Bring, scho

Decision: Dyysno, vicious power of 5, 4 and 3 poduchiti

The subordinate powers of the scholars are linked with the understanding of the minor and the algebraic supplement.

minor deyako element aij visitor n- th order to be called a visitor n- the 1st order, the removal of the row from the outward path is a hundred percent, on the overturning of which there is a vibrating element. signify mij

algebraic additions element aij the visitor is called a yogo minor, taking a plus sign, yaksho sum i + j couples number, і zі with the sign "minus", as the sum is unpaired. signify Aij:

power 7 ("The distribution of the viznachnika for the elements of the deyakiy number"). Business card holder for additional information.

Multiply the matrix by the number, you need to multiply the skin element of the matrix by the whole number.

Slidstvo. The leading multiplier of all elements in the matrix can be attributed to the sign of the matrix.

For example ,.

It can be seen that there are additional matrices, multiple matrices for the number of analogous steps above the numbers. Multiplying matrices is a specific operation.

Tvir two matrices.

Not every matrix can be multiplied. Tvir two matrices Aі V in the designated order AB You can only do it, if the number of hundred of the first multiplier A according to the number of rows in another multiplier V.

For example ,.

Matrix size A 33, matrix size V 23. Tvir AB unhappy, tvir VA it is possible.

Twir two matrices A and B є the third matrix C, element C ij which paths sum of paired additions of elements of the i-th row of the first multiplier and the j-one hundredth of another multiplier.

It is shown in Booleau that there can be solid matrices in the given view. VA

There are three rules for showing two matrices to the creation, but also two matrices in the zalous vipad are not subject to the displacement law, so that AB? VA... Yaksho in okremomu vipad to appear, scho AB = BA, then such matrices are called permutations or commutative.

In the matrix algebra TWIR, two matrices can be a zero matrix, if the matrix of factors is not zero, in contrast to the algebraic algebras.

For example, we know tvir matrices AB, yaksho

Multiplication of matrixes is possible. You can multiply matrices A, V i tvir qix matrix can be multiplied by matrix Z, Then you can write tvir ( AB) Zі A(Sun). In such a case, there is a lack of associative law of multiplication ( AB) Z = A(Sun).