How do you know the algebraic complement of matrices. How to calculate the calculation (determinant) of the matrix? Minor and algebraic addition. Addendum of the algebra $A_(ij)$ of the element $a_(ij)$

Task 1.

For this signatory

know the minors and algebraic additions of elements α 12 , α 32 . Calculate the winner : a) reciting yoga behind the elements of the first row and that of the other row; b) taking off the zeros in front of the first row.

We know:

M 12 =
= –8–16+6+12+4–16 = –18,

M 32 =
= –12+12–12–8 = –20.

Algebraic supplementary elements a 12 and 32 are equally equal:

A 12 \u003d (-1) 1 +2 M 12 \u003d - (-18) \u003d 18,

A 32 \u003d (-1) 3 +2 M 32 \u003d - (-20) \u003d 20.

a) Calculate the vyznachnik, having recited yoga behind the elements of the first row:

A 11 A 11 + a 12 A 12 + a 13 A 13 + a 14 A 14 = -3
–2 +

1
= – 3(8 + 2 + 4 – 4) – 2(– 8 – 16 + 6 + 12 + 4 – 16) + (16 – 12 – – 4 + 32) = 38;

Let's lay out the primary behind the elements of another column:

= – 2 – 2
+ 1
= – 2(– 8 + 6 – 16 + + 12 + 4 – 16) – 2(12 + 6 – 6 – 16) + (– 6 + 16 – 12 – 4) = 38;

b) Calculate by taking off the zeros in front of the first row. Vikoristovuєmo vіdpovіdnu pravіvіst vyznánіvіv. Multiply the third step of the altar by 3 and add to the first, then multiply by -2 and add to the next. Then, in the first row, all elements, except for one, will be zeros. It is possible to deduct such a rank from the vyznachnik for the elements of the first row and countably yogo:

= =
=
=
=

= – (– 56 + 18) = 38.

(The third-order ruler was stripped of zeros in the first column for himself, which is more for the power of the chiefs.) ◄

Task 2.

Given a system of linear inhomogeneous equalizations of algebra

Verify, chi spіlna tsya system, and razі splnostі virishity її: a) according to Cramer's formulas; b) for an additional serum matrix (matrix method); c) Gaussian method.

The split of the whole system is verifiable by the Kronecker-Capelli theorem. For the help of elementary transformations, we know the rank of the matrix

A =

given system and rank of the extended matrix

At =

.

For this, we multiply the first row of the matrix by -2 and store it from the other, then we multiply the first row by -3 and store it from the third, remember the other row and the third column. Take away

At =

~

~
.

Father, rang A= rank At= 3 (there are no numbers). Otzhe, vyhіdna system spilna that maє єdine solution.

a) Behind Cramer's formulas

x= x / , y = y/ , z = z/ ,

=
= – 16;

x =
= 64;

y =
= – 16;

z=
= 32,

we know: x = 64/(– 16) = – 4, y = – 16/(– 16) = 1, z = 32/(– 16)= – 2;

b) For the value of the solution of the system behind the additional pivot matrix, we write down the system of equations for the matrix form AH = . System solutions for matrix forms can be seen x = A –1 . Behind the formula we know the inversion matrix A –1 (Vona іsnuє, so yak = det A = – 16 ≠ 0):

A 11 =
= – 15, A 21 = –
= 16, A 31 =
= – 11,

A 12 = –
= – 3, A 22 =
= 0, A 32 = –
= 1,

A 13 =
= – 14, A 23 = –
= 16, A 33 =
= – 6,

A –1 =

.

System solutions:

X = =
=
=

.

Otzhe, x = –4, y = 1, z = –2;

c) Verify the system by the Gauss method. turn off x from another that third equal. For the first equal, we multiply by 2 and we see the other, then we multiply the first equal by 3 and we see the third:

Z otrimanoї system is known x = – 4, y = 1, z = –2. ◄

Task 5.

The tops of the pyramid are in points A(2; 3; 4), B(4; 7; 3), C(1; 2; 2)і D(-2; 0; -1). Calculate: a) the area of ​​the face ABC; b) the area of ​​the cut to pass through the middle of the ribs AB, AC, AD; c) obsyag pyramids ABCD.

A) It seems that S ABC =
. We know:
= (2; 4; – 1) ,

= (– 1; – 1; – 2) ,

=
= – 9 i + 5 j + 2 k.

Remaining maєmo:

S ABC =
=
;

b) Middle of the ribs AB, NDі AD be at the points Up to (3; 5; 3.5),

M (1.5; 2.5; 3),N (0; 1,5; 1,5) . Dali maєmo:

S sich =
,
= (– 1,5; – 2,5; – 0,5),
= (– 3; – 3,5; – 2),

=
= 3.25i - 1.5j - 2.25k,

S sich =
=
;

c) Oscilki V benquet =
,
= (– 4; – 3; – 5),

=
= 11, That V = 11/6 . ◄

Head office 6

Force F = (2; 3;– 5) added to the point A(1; - 2; 2). Calculate: a) the work of the force F at times, if the point її zastosuvannya, collapsing rectilinearly, moving from the position A at the camp B(1; 4; 0); b) modulus of moment of force F shodo points At.

A) so yes A =F · s , s =
= (0; 6; – 2) ,

That F = 2 0 + 3 6 + (-5) (-2) = 28; A = 28;

b) Moment of force M =
,
= (0; – 6; 2) ,

=
= 24 i + 4 j + 12 k .

Otzhe, =
= 4
. ◄

Manager 8.

Vіdomi peaks O(0; 0),A(– 2; 0) parallelogram SLADі point of intersection of yogo diagonals B(2;–2). Write down the alignment of the sides of the parallelogram.

Rivnyanya side OA you can write in a sentence: y = 0 . Dali, oskelki dot Atє the middle of the diagonal AD(Fig. 1), then following the formulas below the top, you can calculate the coordinates of the vertex D(x; y) :

2 =
, –2 =
,

stars x = 6 , y = –4 .

Now you can know the equal decision of the parties. Vahovuyuchi parallelism of the parties OA і CD, we fold the sides CD: y = –4 . Rivnyanya side OD folded over two vіdomimi points:

=
,

stars y = – x, 2 x + 3 y = 0 .

Zreshtoy, we know the equal side AC looking back at the fact that there is no way to pass through a visible point A (-2; 0) parallel to the straight line OD:

y – 0 = – (x + 2) or 2 x + 3 y + 4 = 0 . ◄

Manager 9.

Dano tops of trikutnik ABC: A(4; 3), B(– 3; – 3), C(2; 7) . Know:

a) equal parties AB;

b) equal height CH;

c) equalization of the median AM;

d) point N peretina median AM that visoti CH;

e) straight line to pass through the top C parallel to the side AB;

f) stand out from the point C to the straight line AB.

A) speeding up the jealousy straight line that passes through two points, otrimaemo equal sides AB:

=
,

stars 6(x – 4) = 7(y – 3) or 6 x – 7 y – 3 = 0 ;

b) Vіdpovіdno to vvnyannya

y = kx + b (k = tg α ) ,

cut coefficient direct AB k 1 =6/7 . Z urahuvannyam understand the perpendicularity of the lines ABі CH kutovy coefficient of height CH k 2 = –7/6 (k 1∙ k 2 = –1). By point C(2; 7) and the cut coefficient k 2 = –7/6 we add height CH: (y – y 0 = k(x – x 0 ) )

y – 7 = – (x – 2) or 7 x + 6 y – 56 = 0 ;

c) Coordinates are known behind the given formulas x, y middle M vіdrіzka BC:

x = (– 3 + 2)/2 = –1/2, y = (– 3 + 7)/2 = 2.

Now for two vіdomih points Aі M we add equal median AM:

=
or 2 x – 9 y + 19 = 0 ;

d) For the value of the coordinates of a point N peretina median AM that visoti CH fold the equalization system

Virishyuchi її, otrimuєmo N (26/5; 49/15) ;

e) So it’s straight, so you can pass through the top C, parallel to the side AB, then their cut coefficients are equal k 1 =6/7 . Todі, vіdpovіdno to vvnyannya:

y – y 0 = k(x – x 0 ) , behind the dot C and the cut coefficient k 1 fold straight lines CD:

y – 7 = (x – 2) or 6 x – 7 y + 37 = 0 ;

f) Walk into the dot C to the straight line AB calculate according to the given formula:

d = | CH| =

Razvyazannya tsієї tasksі is illustrated in fig. 2◄

Manager 10.

Dano chotiri points A 1 (4; 7; 8), A 2 (– 1; 13; 0), A 3 (2; 4; 9), A 4 (1; 8; 9) . Fold the line:

a) flat A 1 A 2 A 3 ; b) straight A 1 A 2 ;

c) straight A 4 M, perpendicular to the plane A 1 A 2 A 3 ;

d) straight A 4 N, parallel to the straight line A 1 A 2 .

Calculate:

e) sine kuta mizh straight A 1 A 4 that flat A 1 A 2 A 3 ;

f) cosine kuta mizh coordinate plane Proxy that flat A 1 A 2 A 3 .

A) vikoristovuyuchi formula alignment of the plane behind three points, we add equal area A 1 A 2 A 3 :

stars 6x - 7y - 9z + 97 = 0;

b) Vrakhovuychi alignment of a straight line that passes through two points, alignment of straight lines A 1 A 2 can be recorded at the sight

=
=
;

c) Z understand the perpendicularity of the straight line A 4 M that flat A 1 A 2 A 3 sled, scho yak direct vector direct s can you take a normal vector n = (6; – 7; – 9) flats A 1 A 2 A 3 . Todi straight line A 4 M s urakhuvannyam canonical direct sign up at the sight

=
=
;

d) So straight A 4 N parallel to straight A 1 A 2 , then their direct vectors s 1 і s 2 can be entered as such that they avoid: s 1 =s 2 = (5; – 6; 8) . Father, straight line A 4 N may look

=
=
;

e) Behind the formula the magnitude of the kuta between the straight line and the flat

sin φ =

f) Valid to the formula of significance Kuta values ​​between flats

cos phi =
=
◄

Manager 11.

Fold flat planes to pass through points M(4; 3; 1) і

N(– 2; 0; – 1) parallel line through points A(1; 1; – 1) і

B(– 3; 1; 0).

Zgidno with the formula alignment of straight lines in space, to pass through two points, straight lines AB may look

=
=
.

Like a plane passing through a point M(4; 3; 1) , then її equal can be recorded at a glance A(x – 4) + B(y – 3) + C(z – 1) = 0 . So how can a plane pass through a point N(– 2; 0; – 1) , then win the mind

A(-2-4) + B(0-3) + C(-1-1) = 0 or 6A + 3B + 2C = 0.

Oskіlki need a plane parallel to the known straight line AB, then with the formulas understand the parallelism of straight lines and planes maybe:

–4A + 0B + 1C = 0 or 4A-C=0.

Virishyuchi system

we know that C = 4 A, B = – A. Let's take the value Wі B in the level of the shukano area, maybe

A(x - 4) - A(y - 3) + 4A(z - 1) = 0.

so yak A ≠ 0 , then otrimane equal is equivalent to equal

3(x - 4) - 14(y - 3) + 12(z - 1) = 0. ◄

Manager 12.

Know coordinates x 2 , y 2 , z 2 specks M 2 , symmetric point M 1 (6; – 4; – 2) shodo flat x + y + z – 3 = 0 .

Let's write the parametric alignment of the straight line M 1 M 2 , perpendicular to the given plane: x = 6 + t, y = – 4 + t, z = – 2 + t. We know t = 1 i, otzhe, dot M tulle straight M 1 M 2 with a flat area: M (7; – 3; – 1) . Bo dot Mє middle vіrіzka M 1 M 2 , then vіrnі rіvnostі.; c) parabolas that make directrix b

Elements of linear algebra are included to the th division of the main types of tasks, as they are seen in the topics of "Linear algebra": calculation of variables, di n

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Square matrix know A) minor element; b) algebraic additional element; V) ... know A) minor element; b) algebraic additional element; c) її vyznachnik, otrimavshi front zeros in the first row. Solution a) Minor element ...

I. elements of linear algebra and analytical geometry

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... element matrices". Appointment. Algebraic supplementary element aik matrix A is called minorМік цієї matrices, multiplications by (-1)і+k: Algebraic additional element... method. Example 1. Given a matrix Know det A. Solution. Let's remake...

Solution: when adding two matrices to the skin element of the first matrix, it is necessary to add another matrix element

Solution

State; name minor element. Todi for appointments is important (1) - algebraic additional element then (2) ... Linear operations on matrices Know the sum of matrices and tvir ... is spilna, then it is necessary knowїї zagalne solution. ...

Methodological recommendations for the study of independent work of the student for the lesson "Mathematics" for the specialty

Methodical recommendations

Such a scapegoat is called minor element aij. Appointed minor- Mij. Butt: Know minor element a12 vyznachnika For ... one lower than that minor dorivnyuє: Algebraic supplementary element the primate is called yoga minor uzyat zі svoїm ...

Minor M ij element aij vyznachnik n -th order is called the signifier of the order ( n-1 ), subtracting from the given primate to the resurrection of the row of that stovptsya, in which the whole element is found ( i th row j th stage).

Algebraic additions element aij asks viraz:

The signatories are in order n>3 are calculated for the help of the theorem about the distribution of the variable for the elements of the abost row:

Theorem. The signator of the richest sum of creative elements, whether there is a line of purity on the corresponding elements of algebraic additions, tobto.

butt.

Calculate the vyznachnik, having announced yoga for the elements of a row of cleaning:

Solution

1. Even though in one row or one column there is only one element that looks like zero, then there is no need to change the signifier. In a different way, the first step is to establish the theorem about the laying out of the chief, we can remake it, the crowning of such power: to add the elements of the next row (stow) to the elements of the row (st), multiplied by the full multiplier, the value of the sign is not.

From the elements of row 3, we can see the second elements of row 2.

From the elements of column 4 we can see the essential elements of column 3 multiplied by 2.

We place the vyznachnik behind the elements of the third row

2. The 3rd order deviant can be assigned to the trikutnik rule or to the Sarrus rule (divine). However, the elements of the vyznachnik are to be finished with great numbers, to which we lay out the vyznachnik, having turned yogo in front:

From the elements of another row, we can see the same elements of the first row, multiplied by 3.

From the elements of the first row, we can see the second elements of the third row.

Before the elements of row 1, we add additional elements of row 2

The arbitrator from the zero row is 0.

Otzhe, forerunners of order n>3 are calculated:

· Reformation of the vyznachnik to a tricot looking after the help of the authorities of the vyznachniks;

· Razkladannyam vyznachnika on elements termіnі abo stovptsya, tim themselves nizhuyuchi yogo order.

Matrix rank.

Matrix rank is an important numerical characteristic. The most characteristic task, which depends on the importance of the rank of the matrix, is the reverification of the summability of the system of linear equations of algebra.

Let's take a matrix A order p x n . Come on k - Dejake is a natural number that does not outweigh the smallest number p і n , tobto,

Minor k-th order matrices A is called the sign of the square matrix order k x k , composed of matrix elements A , yakі know in the back of the line k rows that k stovptsyakh, moreover, the expansion of elements in the matrix A be saved.

Let's look at the matrix:

Let's write down the number of minors of the first order of the matrix. For example, as we choose the third row and another matrix row A , Then our choice is given to the minor of the first order det(-4)=-4. In other words, for the removal of this minor, the first and second rows were added, as well as the first, third and fourth columns of the matrix. A , and from the element that was left over, they folded the vyznachnik.

In this order, the minors of the first order of the matrix are the elements of the matrix.

Let's show the sprat in a different order. We select two rows and two columns. For example, take the first and other rows, and the third and fourth rows. For such a choice, there is a minor in a different order
.

In the lower minor of another order of the matrix Aє minor

Similarly, you can find the minors of the third order of the matrix A . So like the matrix A there are three rows in total, then we select all of them. If you choose three first rows before these rows, then take the minor of the third order:

Іnshim minor of the third order є:

For this matrix A in the order of the third no more than the third, shards

Skіlki w іsnuє minorіv k -Wow matrix order A order p x n ? Chimalo!

Number of minors in order k can be calculated using the formula:

Matrix rank the greatest order of the minor of the matrix is ​​called, the most significant of which is zero.

Matrix rank A signify yak rang(A). By assigning the rank of the matrix and the minor of the matrix, it is possible to create vysnovok, so that the rank of the zero matrix is ​​equal to zero, and the rank of the non-zero matrix is ​​not less than one.

Also, the first method of determining the rank of the matrix є brute force method . This way is based on the assigned rank of the matrix.

Let us know the rank of the matrix A order p x n .

If you want one element of the matrix that looks like zero, then the rank of the matrix is ​​at least more than one (shards are minor of the first order, not equal to zero).

Let's move on to minors in a different order. If all minors of a different order are equal to zero, then the rank of the matrix is ​​equal to one. If we want one non-zero minor of a different order, we pass to the enumeration of minors of the third order, and the rank of the matrix is ​​at least two.

Similarly, if all minors of the third order are equal to zero, then the rank of the matrix is ​​equal to two. If we want one minor of the third order, if we look at zero, then the rank of the matrix is ​​at least three, and we pass to the enumeration of minors of the fourth order.

Significantly, the rank of a matrix cannot exceed the smallest number p і n .

butt.

Find the rank of a matrix
.

Solution.

1. Since the matrix is ​​non-zero, її rank is not less than one.

2. One of the minors in another order
vіdminny vіd zero, otzhe, matrix rank A not less than two.

3. Minor in the third order

Usі minors of the third order equal to zero. Therefore, the rank of the matrix is ​​double.

rank(A) = 2.

Establish other methods for determining the rank of the matrix, which will allow you to take the result in less computational work.

One of these methods is method of oblyamіvnyh minorіv . When using this method, the calculation is very fast, and yet the stench is cumbersome.

There is one more way to determine the rank of the matrix - for the help of elementary transformations (Gauss method).

The next transformation of the matrix is ​​called elementary :

· Permutation of the rows (or stovptsiv) of the matrix;

multiplication of all elements of any row (stowptsya) of the matrix by a certain number k, vіdmіnne vіd zero;

Addendum to the elements of any row (stovptsya) of the same elements of the next row (stovptsya) of the matrix, multiplied by a certain number k.

The matrix is ​​called the equivalent matrix A, yakscho At otrimana s A for the help of the final number of elementary transformations. The equivalence of matrices is indicated by the symbol « ~ » to sign up A~B.

Knowing the rank of the matrix for the help of elementary transformations of the matrix is ​​grounded on hardened: like a matrix At taken from the matrix A help the last number of elementary transformations, then r ang(A) = rank(B) , then. ranks of equivalent matrices equal .

The essence of the method of elementary transformations is based on the given matrix, the rank of which we need to know, to trapezium-like (in a clear slope to the upper tricot) for the help of elementary transformations.

The rank of matrices of this kind is easy to know. Vіn dorivnyuє kіlkostі rowkіv, scho vengeance b one non-zero element. Since the rank of the matrix for the hour of the elementary transformations is not changed, then the value will be the rank of the output matrix.

butt.

Using the method of elementary transformations, find the rank of a matrix

.

Solution.

1. Remember the first and other rows of the matrix A , so as an element a 11 = 0, and the element a 21 vіdminny vіd zero:

~

In the opposite matrix, the element is the most common one. In the next step, it was necessary to multiply the elements of the first row by . Zrobimo all the elements of the first column, crim first, zero. In the other row, zero is already є, until the third row, dodamo is first, multiplied by 2:

The element of the removed matrix is ​​considered to be zero. Multiply the elements of another row by

Another stovpets otrimanoї matrix may be necessary, the shards of the element are already equal to zero.

so yak , A , then we remember the third and fourth columns by the months and multiply the third row of the taken matrix by:

The external matrix is ​​brought to a trapezoid-like, її rank more equal number of rows, which wants to eliminate one non-zero element. There are three such rows, and the rank of the external matrix is ​​three. r ang(A)=3.

reversal matrix.

Let me have the matrix A .

Matrix, pivot matrix A , called the matrix A-1 so what A -1 A = A A -1 = E .

The reverse matrix can be used only for a square matrix. Moreover, she herself is the very expansiveness, like a free matrix.

In order for the square matrix to be small, it is guilty of being unvirgin (tobto. Δ ≠0 ). Tsya umova є i sufficient for іsnuvannya A-1 to matrix A . Otzhe, whether a non-virogene matrix can be reversed, and united.

Algorithm for the significance of the gate matrix on the application of the matrix A :

1. We know the sign of the matrix. Yakscho Δ ≠0 , then the matrix A-1 Ісnuє.

2. We add the matrix to the complementary algebra of elements in the output matrix A . Tobto. at the matrix At element i - th row that j - stovptsia will be algebraic additions A ij element aij output matrix.

3. Transpose the matrix At and taken B t .

4. We know the inversion matrix by multiplying the otriman matrix B t per number .

butt.

For this matrix, know the reverse and vice versa:

Solution

We quickly describe the algorithm of the significance of the gate matrix.

1. To calculate the basis of the pivot matrix, it is necessary to calculate the origin of this matrix. Speeding up with the rule of tricks:

Matrix є nevirodzhenoyu, otzhe, won the werewolf.

We know the additions of the algebra of all elements of the matrix:

From the knowledge of algebraic additions, a matrix is ​​formed:

that transpose

After dividing the skin element of the removed matrix into the vyznachnik, we take the matrix back to the exit:

The re-verification is due to the multiples of the obtained matrix on the way out. If the reverse matrix is ​​found correctly, the result of the multiplication will be a single matrix.

For the significance of the reversal matrix for this, you can speed it up using the Gauss method (indeed, it is necessary to move forward, because the matrix is ​​inverted), which I use for independent work.

Minori matrices

Come on, it's square matrix A, nth order. Minor deyago element a ij, matrix n -th order is called vyznachnik(n - 1) - th order, deleting from the wake of the weekend row that column, on the front of which there is an element ij. It is designated M ij.

Let's look at the butt matrix 3 - yoga in order:

Todi zgidno s appointed minor, minor M 12 vyznachnik:

With whom, for help minors you can ease the billing matrix. It is necessary to spread matrix in a row and then vyznachnik add the sum of all the elements of the row to their minors. Unfolding matrix 3 - see the order in the following way:

The sign in front of the creation is old (-1) n de n \u003d i + j.

Algebraic additions:

Algebraic additions element a ij is called yoga minor, taking зі with the "+" sign, i.e. the sum of (i + j) is a paired number, i зі with the sign "-", i.e., the sum is an unpaired number. It is designated A ij. A ij \u003d (-1) i + j × M ij.

Todi it is possible to reformulate into a greater power. Significant matrix add more elements to a certain row (rows or rows) matrices on a daily basis algebraic additions. Butt:

4. The reverse matrix and її calculation.

Let A be square matrix nth order.

Square matrix And they are called non-virgin, like matrix(Δ = det A) is not equal to zero (Δ = det A ≠ 0). In the other fall (Δ = 0) matrix And it is called a virogen.

Matrix, allied to matrices Ah, it's called matrix

De A ij - algebraic extension element a ij given matrices(it shows up just like that, like i algebraic extension element matrix).

matrix A -1 is called serum matrix A, so that the mind is victorious: A A - 1 \u003d A -1 A \u003d E, de E - single matrix in the same order as matrix A. matrix A -1 maє sami razmіri, yak i matrix A.

reversal matrix

How to use square matrices X i A, which pleases the mind: X A \u003d A X X \u003d E, de E - single matrix in the same order, then matrix X is called the gate matrix to matrix A i is assigned A-1. Be-yaka non-virogene matrix may reversal matrix and before that, only one, so for that, schob is square matrix A is small reversal matrix, necessary and sufficient, schob її vyznachnik letter vіdminny vіd zero.

For otrimanna serum matrix vikoristovuyut formula:

De m ji dodatkovy minor element a ji matrices A.

5. Matrix rank. Calculation of the rank with the help of elementary transformations.

Let's look at a rectilinear mxn matrix. We can see that in the matrix there are k rows and k columns, 1 £ k £ min (m, n) . From the elements that stand on the peretina of the seen rows and stovptsiv, we fold the k-th order. Mustaches are called the minors of the matrix. For example, for a matrix, you can fold the minors in a different order ta minor of the first order 1, 0, -1, 2, 4, 3.

Appointment. The rank of a matrix is ​​called the highest order of the top view of the zero of the minor of the matrix. Designate the rank of the matrix r(A).

At the pointed butt, the rank of the matrix is ​​equal to two, shards, for example, minor

The rank of the matrix is ​​manually calculated by the method of elementary transformations. Before the elementary transformations, one should add the following:

1) permutations of rows (stovptsiv);

2) multiplying a row (stovptsya) by a number, not counting zero;

3) adding to the elements of a row (stovptsya) the corresponding elements of the next row (stovptsya), forward multiplying the number by the day.

Qi transformations do not change the rank of the matrix, that is, 1) when rearranging the rows, the alternator changes the sign of i, if the wine is not equal to zero, then I will not become; 2) when multiplying the row of the arbitrator by a number that is not equal to zero, the arbitrator is multiplied by the whole number; 3) the third elementary transformation in the beginning changes the signifier. In this way, viroblyayuchee over the matrix of elementary transformation, you can take the matrix, for which it is easy to calculate its rank i, hence, the exit matrix.

Appointment. The matrix, taken from the matrix for additional elementary transformations, is called equivalent and is designated A At.

Theorem. The rank of the matrix does not change with elementary transformations of the matrix.

For the help of elementary transformations, you can bring the matrix to the so-called stepwise look, if the calculation of the rank is not important.

matrix it is called a step, as it may look out:

It is obvious that the rank step matrix equal to the number of non-zero rows , because є minor order, not equal to zero:

.

butt. Designate the rank of the matrix for the help of elementary transformations.

The rank of the matrix is ​​equal to the number of non-zero rows, tobto. .

Algebraic additions- Understanding matrix algebra; the hundredth element of the element aij of the square matrix A becomes the multiplication of the minor of the element aij by (1)i+j; Aij is assigned: Aij=(1)i+jMij, de Mij minor element aij of matrix A=, then. forerunner... ... Economic and Mathematical Dictionary

algebraic extension- understand matrix algebra; the hundredth element of the element aij of the square matrix A becomes the multiplication of the minor of the element aij by (1)i+j; Aij is assigned: Aij=(1)i+jMij, de Mij minor element aij of matrix A=, then. matrix arbiter, ... ... Dovіdnik technical translation

Div at st. Vyznachnik… Great Radianska Encyclopedia

For the minor M, the number that is equal to the M minor to the order of k, the rows with the numbers and the columns with the numbers of the square matrix A of the order n; the matrix of the order n k, taken from the matrix Avikresluvannyam rows and columns of the minor M; Mathematical Encyclopedia

Wiktionary has the article “additional” Addendum can mean ... Wikipedia

The operation, until heaven, set the value of the multiplicity of Mdan's multiplier X, if not, if we already know Mі N, then in another way we can renew the impersonal X. It is important that the impersonal X is endowed with a structure, ... Mathematical Encyclopedia

Abo is a determinant, in mathematics the record of numbers is like a square table; Even more often, under the understanding of the vyznachnik, the vyznachnik may be on the uvazi as the meaning of the vyznachnik, so the form of his record. Collier Encyclopedia

About the theorem from the theory of immovability of divs. article The local theorem of Moivre Laplace. Laplace's theorem is one of the theorems linear algebra. Named after the French mathematician P'er Simon Laplace (1749–1827), who is credited with the formula ... ... Wikipedia

- (Laplacian matrix) one of the graph behind the help matrix. The Kirchhoff matrix is ​​victorious for the skeleton of a given graph (matrix theorem about a tree), and also victorious in the spectral theory of graphs. Zmist 1 ... ... Wikipedia

Equalities are called mathematical parallelism, which demonstrates the equality of two parallels in algebra. If equivalence is fair for any acceptable meanings of the unknown, which enter before the new, then it is called the sameness; for example, mindfulness ... Collier Encyclopedia

Books

• Discrete Mathematics, A. V. Chashkin. 352 pages The handbook consists of 17 divisions from the main divisions of discrete mathematics: combinatorial analysis, graph theory, boolean functions, folding calculation and theory coding Revenge...

vyznnik behind the elements of the row

Further power is connected with the understanding of the minor and algebraic additions

Appointment. Minor an element is called a viznnik, warehouses of elements that were lost after Sundayioh drains tajth stovptsya, on the peretina yakikh є tsey element. Minor of the primordial element n-th order may order ( n- 1). Let's designate yoga through.

example 1. Come on also .

Cey minor to go out of the A way of the Sunday of the other row and the third stovptsya.

Appointment. Algebraic additions element is called vіdpovіdny minor, multiplications nat. , dei-Row number ij-Stovptsya, on the peretina there is a given element.

VІІІ. (The laying out of the vyznachnik for the elements of the third row). The signator of the richest sum of creations of the elements of a certain row of algebraic additions.

.

butt 2. Let me go

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example 3. We know the primate of the matrix, reciting yoga behind the elements of the first row.

Formally, the theorem of that іnshi authority of the vyznachnikіv zastosovnі is still only for the vyznachnіv matrices not higher than the third order, the fragments of the іnshі vyznachniki were not considered by us. The advent of the appointment will allow you to expand the power of the rulers, no matter what order.

Appointment. Vyznachnik matrices A The n-th order is called the number, calculated for the additional successive calculation of the theorem about the distribution of those other powers of the rulers.

It is possible to believe that the result of the calculation cannot be stale, since in such a sequence for some rows and stovptsiv there will be higher-ranking authorities. Vyznachnik for the help of his appointment is unambiguous.

If you don't want to avenge the explicit formula for the value of the higher order, you do not allow you to follow the path to the lower order matrices. Such a designation is called recurrent.

butt 4. Calculate the vyznachnik: .

If you want the theorem about the layout can be zastosovuvat to any row or a given matrix, it is less to calculate the amount of space when arranging according to the column, in order to avenge as much as possible zeros.

If the matrix does not have zero elements, then we take them away for additional power 7). Multiply the first row sequentially by the numbers (–5), (–3) і (–2) і dodamo yogo to the 2nd, 3rd and 4th rows і subtract:

We lay out the vznachnik, scho viyshov, according to the first step and take it:

(fault from the 1st row (-4), from the 2nd - (-2), from the 3rd - (-1) from the authorities 4)

(Oskіlki vyznachnik avenge two proportional stovptsі).

§ 1.3. Deyakі see the matrix and їх vyzniki

Appointment. Square m an matrix, which is lower or higher behind the head diagonal, have zero elements(=0 when i j, or =0 at i j) calledknitted .

Їх is schematic budova vіdpovіdno may look: or .

Here 0 means zero elements, and more elements.

Theorem. The signifier of a square tricot matrix is ​​a rich addition to the її elements, so as to stand on the head diagonal, tobto.

.

For example:

.

Appointment. A square matrix, in which position the head diagonal has zero elements, is calleddiagonal .

Її schematic view:

Diagonal matrix, which has fewer single elements on the head diagonal, is called solitary matrix. Vaughn is indicated through:

The signifier of the single matrix is ​​more than 1 tobto. E=1.