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0INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE "RAPID DEVELOPMENT OF MULTILATERAL INTERNATIONAL RELATIONS FROM A
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UDC 372.8:51
METHODS FOR CALCULATING DETERMINANTS OF SQUARE MATRICES AND THEIR APPLICATION
JIbragimov.R, 2Yermatov.Sh
1doctor of pedagogical science, professor South Kazakhstan Pedagogical University named after U. Zhanibekov, Shymkent, 2lecturer South Kazakhstan Pedagogical University named after U.
Zhanibekov, Shymkent https://doi.org/10.5281/zenodo.11518440
Abstract. In this article, we will consider the solution of indefinite equations and an algorithm for finding the largest common divisors of several numbers using methods for calculating the determinants of non-square matrices. Matrix terms and determinants are often found in the algebra course. First of all, square matrices and their determinants are considered. However, the scientific literature does not consider non-square matrices and calculations of the values of their determinants.We introduced a new concept of "non-square matrices and methods for calculating their determinants" (we called them calculators), formulas for calculating the determinants of non-square matrices. Because of their abundance, we have tried to identify each species separately andprovide methods for calculating their value. Since there are many practical mathematical problems that need to be solved by finding the value in calculators, we have touched on only a few of them. Special attention was paid to the problems of solving indefinite equations using calculators of type 1xn, finding the largest common divisors of several numbers.
Keywords: non-square determinants, a method for calculating their value, solving indefinite equations, an algorithm for finding the largest common divisors of several numbers.
Analysis and research methods
For the solution of indefinite equations, the work of Diophantus of Alexandria is the basics. Diophantus of Alexandria showed solutions of indefinite equations and systems of equations by various artificial methods, but there is still no general solution method [1,141 — 153p. ].
In the works of famous scientists Z.I.Borevich [2], M.Orazbaev [3], I.M.Vinogradov [4], A.G.Kurosh [5], D.K.Faddeev [6], the problems of matrices and determinants are considered. In the works of the above-mentioned scientists, square matrices and their determinants are considered most of all. However, recently some scientists have begun to consider non-square matrices and calculate the values of their determinants. For example, H.Mink [7] (he called such matrices nonsquare matrices or permanent) considered non-square matrices and calculations of their determinants, but his method is not suitable. A lot of scientists were engaged in solving indefinite equations: A.O.Gelfond [8], V.O.Sarpinsky [9], P.L.Chebyshev [10], M.B.Gelfand and I.S.Pavlovich [11], I.J.Depman [12], T.A.Lavrinenko [13], L.Y.Kulikov [14], A.I.Kostrikin [15], L.Ya.Okunev [16], T.A.Lavrinenko [17], I.G.Bashmakova [18], I.G.Bashmakova [19], A.O.Gelfond [20], A.O.Gelfond [21], Davenport [22], A.Salihu, F.Marevci [23].
In this paper, we considered new methods for solving indefinite equations using raskulants. We called the determinants of non-square matrices raskulants m x n (m ^ n), there are many types of them.
There is no definition of the values of the figures in the scientific literature (methods for calculating the values of non-square determinants). First, we consider the types of raskurants of 1xn , and then, using the calculation of the values of raskulants, we will solve indefinite equations.
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Definition. A raskulant (non-square determinants) is a table consisting of m rows and n columns, which has the form (m ^ n ):
an a12 ... a1n
«21 Ü22 ... a2n (11)
a ml am2 . .. « mn
where a£y (i=1,2,...,m; j=1,2,...,n) they are called the elements of the raskulant, the first index m- line number, and n- the column number. Then the raskulant (1.1) is denoted by its elements by |a£y |. Raskulants are indicated by large Latin letters A, B, C ... then A=h;|, (i=1,2,...,m; j=1,2,...,n)
And let's agree to call the determinants (determinants) of the type of matrices the mhp raskulants.
Definition: A raskulant consisting of a single line |a1 a2 a3 ... an| it is called a raskulant string / m=l, n=n/.
aii
Definition: A raskulant consisting of a single column
«21 am 1
it is called a decoupling
column / m= m , n=1/.
Definition: Raskulant at m ^ n it is called rectangular, when m > n shortened, when m < n long.
Definition: Raskulants are called mutually typical raskulants if they have equal rows and columns.
Definition: If the corresponding elements are equal for typical raskulants, then they are
equal.
If for the raskulants A=|a£y|, (i=1,2,...,m; j=1,2,...,n) and B=|&£y|, (i=1,2,...,m;
j=1,2, ,n) equal , that A=B,
For non-typical raskulants, the concept of equality is not defined.
In scientific works, the concept of a raskulant and the definition of its meaning do not occur, i.e. such work has not been fully studied.
Therefore, the concept of a raskulant must be given a coherent structure. Due to the variety of raskulants, we tried to define each of them and apply the method.
Raskulants can generally consist of m rows (columns) and from n columns (rows), but
necessarily m ^ n. That is why there is a difference between square determinants and raskulants.
The paper considers the concept of a raskulant, which has the opposite meaning to the square (determinant) determinant and finding the values of the raskulant, i.e. \a b\ , \a b c\ , \a b c d\, \a b c d e\ and the like
Definition: 1. By the decouplants of the numbers a and b , for example |a b | end
we
=a-b (1)
consider the difference a end b , that is |a ¿|=a-b, 1-Conclusion Means |a
Definition: 2. The decouplants of the three numbers a, b, c or |a b c| we understand the following expression:
|a b c^a — b b — c^a-b-fb-c) =a-2b+c (2)
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2- Conclusion Means |a b c|=
Definition: 3. The raskulants of the four numbers a, b, c, d or |a b c d| we understand the following expression:|a b c d| = |a — b ô — c c —d| = |a —2ô + c ô — 2c + d|=a-2b+c-(b-2c+d)=a-3b+3c-d (3)
Now, in these three definitions of raskulants, we will find out the general pattern. For this, the coefficients of the right-hand sides of the equalities (1), (2), (3), we will present it in the following form:
1 -1 1 -2 1 (4). 1 -3 3 -1
From the above triangle, the following alternating pattern can be derived: To find the value
of a row-decoupling (composed of one row and n columns |a1 a2 a3 ... an|) you can use the
following formula: |a. a2 a3 ... an|=C,0a1 — C^a^+...+C:£an (5)
Using this formula, you can find the value of any raskulant composed of a single line.
Definition: 4. Let's call it any layout |a-. a2 a3 ...aJ . . ......
123 ' the value of which is equal to the
value of the sum of the alternating expression: Ql a2+ - an .
Then |ai ... a^=C°ai — C>2+. • +Can (5) the sign alternates here. Above, we
have derived a formula for finding the values of the figures consisting of one row and n columns. Examples of finding the values of raskulants: 115 10|=15-10=5;
1105 21 310|=|84 — 289|=84+289=373;
|24 20 10 28|=|4 10 — 18|=|—6 28|=-6-28=-34;
Now we consider the practical significance of formula (5).
We show the possibilities of solving the following types of problems using formula (5) (finding the values of the 'raskulants consisting of one row and n column):
a) Determination of the roots of indefinite equations of the type
n=0 using raskulants of the type 1xn ;
b) Finding solutions to an indefinite equation of the form:
n =d (1) using a raskulant of the type 1xn Finding the largest common divisor of several numbers: The largest common divisor LCDfa., a2, a3, ... , an). 1. The system of evidence and scientific argumentation
Now we consider each one separately. When solving equations in the form
n=0 (1) It turns
out that it is very convenient to use a 1xn raskulant.
Let's consider several types of such equations and ways to solve these equations: 1.1 If n=2, then equation (1) has the form ax+by=0 (1.1)
ax+by=0 (1.1) To solve such an equation, we will find the value of the following decouplants: x= |0 ¿| =-b, y= |a 0| =a .
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The general solution x = —b + bk , y = a — ak , where k = 0, ±1, ±2 ...
Example 1. 15x+7y= 0
x=|0 7|= -7 y= 115 0|= 15
The general solution to this equation is x=-7+7k, y= 15-15k, where k = 0,±1,±2...
1.2 Using this method, you can find the roots of an equation of the form axn+byp=0 (1.2) Its roots x=\l—b + bk, y=Pila — ak , where k =
0,±1,±2 ...
Example 2. x1201+2y522=0 x1201=|0 2|= -2 , x= 12°ll=2 y522=|1 0|= 1 , y= 52yr
1.3 If n=3 then equation (1) has the form ax+by+cz=0 (1.3) In this case, to find the roots, we use finding the values of the following decouplants.
1-method x= lb cl=b-c, y= —lc al=-(c-a), z= la bl=a-b The general solution of this equation is as above x=(b-c)k, y=(c-a)k, z=(a-b)k . k =1,2,3... Example 1. 1) 6x+3y+2z=0 1-method: x=l3 2l = 1; y = —l6 2l =—4;
z = 16 3 l =3 . Then the general solution is: x=k; y=-4k; z =3k; where k =1,2,3.
1.4 These methods can also be used to find equations of the form axn+byp+czm=0 (1.4)
The roots of such equations have the form: x="¡{b — c)k , y=\j(c — a)k , z="¡{a — b)k .
1.5 If n=4, then equation (1) has the form ax+by+cz+dx=0 (1.5)
In this case, we use the values of the following declutters x= lb c dl=b-2c+d, y= — la c 2dl=2c-a-2d, z= l2a b dl=2a-2b+d, x = —la b cl=2b-a-c . Then the common roots of the equation are (1.5)
x=(b-2c+d)k, y=(2c-a-2d)k, z=(2a-2b+d)k, x =(2b-a-c)k.
In this method, the number of members can be expanded to n= 5,6,7,...
1.6 If n= 5, then the equation under study has the form ax+by+cz+dx+eM=0 (1.6)
To solve such an equation, we use the values of the following decouplants: x= lb c d el=b-3c+3d-e, y= —la c d 3el=3c-a-3d+e, z= l3a b d 3el=3a-3b+3d-3e, x = —l3a b c el=3b-3a-3c+e, m = la b c dl=a-3b+3c-d.
Then the general solution of this equation is ax+by+cz+dx+eM=0 (1.6)
x=(b-3c+3d-e)k, y=(3c-a-3d+e)k, z=(3a-3b+3d-3e)k, x =(3b-3a-3c+e)k, w =(a-3b+3c-d)k. k=1,2,3...
1.7 Hence the roots of the equation axn+bym+czp +dxu=0 (1.7) x= ¡(b — 2c + d)k, y=1^{2c — a — 2d)k , z = ¡{2a — 2b + d )k, x = ¡{2b — a — c)k.
2. Finding a solution to an indefinite equation of the form:
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n =d (1.8) with the help of a raskulant of the form 1xn To solve the equation a1x^1 + a2x^2 + a3Xg3 +...+anx^n =d (1.8), it turns out that raskulants of the type 1xn are very profitable. Let's look at some examples and solve with the help of raskulants of the type 1xn.
2.1 The equation is given ax+by=c (a±b) (1.9) To find the roots of equation (1.9), we use the values of the following raskulants:
1-method: Then the solution of this equation Д=|а Ь| = а — Ь, Дх=|с Ь|=с-
Ь,
Ду = |а с| = а — с. х = Д^ — bfc; У = Д^ + ak.
ас
а —
b
, с ас с
+ - =--a+-
Ь Ь b
2-method: x = |ô c| = ô — c, y= —
Let's look at some examples. Example 1. 54x + 37y = 1
1-method: A= |54 37| = 17, Ax = |1 37| = —36 , Ay = |54 1| = 53 . Then the solution of this equation is x = = — 36 ; ^ = ^ = ^ and the general solution is
x = -—-37fc; y=—+54k.
17 J 17
2-method: x = 137 1| = 36, y=
va 54
54 —
37
,1 54 _ . 1 1943
+ — =--54+— =
37 37 37 37
2.2 To find the roots of the equation ax+by+cz=d (2.0) let's use the values of the following 'raskulants:
1-method: x = |b c| = b — c, y= |c а| = с — а, z= - + |a b| = - +a-b. here, the
с с
calculation is carried out by definition 5.
2-method: When solving equation (2.0), you can use definition 2. To do this, we use the values of the raskulants of the form x = |Ь с d| = b — 2c + d, y= —|а с d| = 2c — a —
d, z=-+|a b d| = -+a-2b+c .
с с
Example 2. 2x+3y+6z=18 x = 3 — 6 = —3, y = 6 — 2 = 4, z = 3 + 2 — 3 = 2
2.3 To find the roots of the equation axm+byn+czp=d (2.1) let's use the values of the
following 'raskulants: х= "//(Ь — с) , у= //(с — a) , z = ^ + (a — b) , a, b, c, d - any numbers.
2.4 To find the roots of the equation ax+by+cz+dx=e (2.2) let's use the values of the following 'raskulants: x= |Ь с d^b-2c+d, у= — |а с d|=2c-a-d,
z= |2а b d^2a-2b+d, т = ^- + Ь + а— c.
2.5 To find the roots of the equation ax+by+cz+dr+eM=m (2.3) let's use the values of the following 'raskulants: x= |Ь с d e|=b-3c+3d-e,
у= —|а с d e^3c-a-3d+e, z= |3а b d e^3a-3b+3d-e, t = —|3a b с e^3b-3a-3c+e,
m . 1 1
ш = —+a-b+c-d.
e
3. Finding the largest common divisor(LCD) of several numbers: LCD(a-i, a2, a3
aj.
Example 1. It is necessary to find LCD(74,36) .
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Solution: To find a LCD(74,36) we calculate the values of the raskulant of the form 174 36|:
174 36|=38.
38 if more 36 then, we calculate the raskulant of the form 138 36|.
138 36|=2. We will stop this process when 36:2 . Then it will be LCD(74,36) =2.
Example 2. We need to find LCD(85, 40) .
Decision: |85 40|=45, ^5 40|=5 40:5
Then then (85,40) =5.
Example 3. Find it LCD(85,24).
Decision: 185 24|=61, ^1 24|=37 ; Here we get a prime number, then the process stops. Meaning LCD(85,24) =1.
Example 4. It is necessary to find LCD (84,124,48) .
Decision: We calculate the values of the raskulant of the form 1124 84 48|
^24 84 48|=|40 36|=4.
We have the smallest number among the data 48. If 48:4 then the process stops. Meaning LCD(84,124,48) =4 , because 48:4=12
Example 5. need to find LCD(85, 35, 27) .
Decision: | 85 35 27|=|50 8 | =42, | 42 27| =15, | 27,15 | =12, |15,12|=3, | 12,3| =9, | 9,3| =6, | 6,3| =3, | 3,3| =0, (85,35,27) =1. Example 6. need to find LCD(18,24,30) .
Decision: | 30 24 18 | = |6 6|, from there 18:6 then LCD(18,24,30) = 6 .
Example 7. need to find LCD (120,80,20) .
Decision: | 120 80 20| = |40 60 | =-20
then LCD(120,80,20) =20 .
Example 8. need to find LCD(78,117,195).
Decision: | 195 117 78| = |78 39 | =39, if 78=39=2 then LCD(78,117,195) = 39. Example 9. need to find LCD(110,154,286) .
Decision: | 286 154 110 | = | 132 44 | =88. In this case, we continue the process.110 88| =22. then LCD (110,154,286) =22.
4. New method: Rules of application in solving a system of indefinite equations by calculating the values of '2x3 raskulants (consisting of two rows, three columns).
\axx + a2y + azz = 0
1) When solving any such systems of indefinite equations as s end
[ bx + ^y + b3z = 0
[a-x + \y + cxz + dj = 0,
\ Kramer's method is not applicable. We asked ourselves this question:
[a2x + by + c2z + d2t = 0.
How should we act in this case? We verbally (quite by accident) introduced new types of 2x3 raskulants (non-square determinants) consisting of two rows, three columns. We calculated the values of the following 2x3 decouplants (non-square determinants), various calculation methods were invented by us as follows:
A =
A, =
a a2 a3 al - a2 a2 - a3
b, b2 b3 K - b2 b2 - b3
0 a2 a3 -a 2 a2 - a3
0 b2 b3 - b2 b2 - b3
= (a1 - a2 Xb2 - b3 )-(b1 - b2 Xa2 - a3 );
= - a2b2 + a2b3 + a2b2 - b2a3 = a2b3 - b2a3;
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Ay =
A, =
aj 0 a
b1 0 b3
a1 a2 0
b1 b2 0
a - a
b - a
= - ab + a3b ;
ax a 2 a 2
=ab 2 -a2h ;
x = ■
bi - b2 b2
Next, the unknowns were found (as in the Kramer method) by the following formulas:
Ax = Ay = AL
A ' ^ A ' Z A '
So, we find a general solution using the following equalities: x = A't, y = AJ • t, z = A ' t Here t: 0,± 1, ± 2 + 3 ...
Now we present the solution of several such systems of equations. \2x + y - 3z = 0 |3x + 3 y - 8z = 0
Such equations are called homogeneous. Any homogeneous systems of indefinite equations can be solved using our methods.
Decision: Let's solve the above system of equations (1) using the method of finding the raskulants (using formula (1), we find the values of non-square determinants with two rows, with three columns).Calculate the values of the following 2x3 raskulants, consisting of two rows, three columns:
Example 1.
(1) Solve the systems of equations.
A =
A y =
2 1 - 3
3 3 - 8
2 0 - 3
3 0 - 8
=11,
=7,
A =
A =
Then, you can find a private solution x =
0 1 - 3
0 3 - 8 2 1 0 3 3 0
1
= 1
=3
11
7 3
y = —, z = — . The general solution 11 11
of the system of equations is found by equalities: x = Axt = t, y = A • t = 7t, z = Azt = 3t Here t: 0,+ 1, + 2,+ 3 ...
i2x + 2y - z = 0
Example 2. |3x - y + 4z = 0 We calculate the following values of non-square
determinants with two rows, with three columns:
A
2 0 -1 3 0 4
=-11 A =
2)
2 2 0 3 -10
ax+by+c\z+dxt = 0,
=-8
x = ■
A
7 -12
2 2 -1 3 -14
y =
■12,
-11
-12
A =
0 2 -1 0 -14
- 8
=7,
z = ■
-12
\a2x + by + ^z + d2t = 0. formulants. x=
(2) To solve such equations, we calculate the following
t= -
C1 d
b2 C2 d2 , y= -
1 C1
2 C2
^^ c^ d ^ a2 C2 d 2
z=
a b dj
a2 ^^2
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Here are examples of solving the following systems of indefinite equations.
Example 3.
\2x + 3y + 5z + m = 0 Ix + 2 y + 4z + 5m = 0
Decision:
x =
3 5 1 2 4 5
=10, y = -
2 5 1 1 - 4 5
2 3 1 2 2 5
15, z = =5, m = -
1 2 5 1 2 4
= 0. The general solution has the form: x - 101
y - -15t, z - 5t, m - 0t, where is any number. [ 3x + y + 4z + 5m - 0 15x + 2y + 6z + 3m - 0
Example 4.
Solution:
Solve the systems of equations.
x =
1 4 5
2 6 3
=-13
y =
3 4 5 5 6 3
=4
3 1 5 5 2 3
=10, m = -
3 1 4 5 2 6
= -1
The general solution: Example 5.
x --131, y - 4t, z - 101, m = -t. where is any number. 3x + 2x + z - k - 3 x - 4y + 2z + 3k - -16 Solve the systems of equations. 2x - y - 3z - 2k - 11 Decision: 1-method
' 3 2
A =
3 2 1 -1
1 - 4 2 3
2 -1 - 3 - 2
=66
A, =
1 -1
-16 - 4 2 3 11 -1 - 3 - 2
= 80
3 3 1 -1 3 2 3 -1
A y = 1 -16 2 3 = 204 A z = 1 - 4 -16 3
2 11 -3 -2 2 -1 11 - 2
3 2 1 3
A * = 1 2 -4 -1 2 -3 -16 11 = 116
80 x = —, 66 y = 204 66 , z _ - 334 66 , k = 116 66 ■
= -334
2-method Calculate the following raskulants
x =
3 1 -1 3 2 -1
y= - 1 23 = -30, z= 1 - 4 3
2 - 3 - 2 2 -1 - 2
2 1 -1
- 4 2 3 = -15,
-1 - 3 -2
3 2 1
k=- 1 - 4 2 =
2 -1 -3
The general solution: x=-15t, y=-30t, z=42t, k=-63t, where is any number. The results of the study;
1) Now we present an algorithm for finding the largest common divisor of several numbers: LCD(ai,a2 ,a3 ,..., an):
z
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1) To find LCD(a!, a2, a3),calculating the raskulants | ax, a2 , a3 | при a3 > a2 > ax. If LCD (ax, a2, a3, a4, a5 ) then we calculate the raskulants | ax a2 a3 a4 a5 | при a5 > a4 > a3 > a2 >
2) Using the formula | ax a2 a3 ... an-1 an |=C,0a1 — C^a2 + C^a^—... C,£-2an-1 + we calculate the values of the raskulants.
When calculating the raskulant, there are two cases:
a) The value of the raskulant is zero. Then the LCD will be equal to the previous value of the raskulant.
b) If the value of the raskulant is equal to one, then the LCD is equal to one.
Here are some examples:
1-the case. For example :LCD (25,12,18)=1 because | 25 18 12| = |7 6 | =1
2-the case. For example LCD (5,3,2)=1 because |5 3 2| = |2 l|=1
2) Effective methods for solving various homogeneous and inhomogeneous indefinite equations using the method of finding the values of the raskulants are presented.
3) Analyzing different ways and methods of solving homogeneous indefinite systems of equations, we came to the following conclusion:
anxx ^ $21X2 ^ a^jX^
= 0 1. Systems of
a) To solve any homogeneous
indefinite equations can be used with formulants consisting of 2 rows and n columns.
a^-^x^ ^21^2 a3lx3 a¿^x^ — b^ b) To solve any heterogeneous problem <a12x1 + a22x2 + a32x3 + a42x4 — b2 2.
aux, I a^x^ I a33x3 I — b^
Systems of indefinite equations, we also use formulas consisting of 3 rows and n columns.
It follows from this that, in any case, it is possible to solve a system of equations consisting of n - unknowns, m - equations, using n-row and m-column type calculators.
1 1 ^3 1 1 ^fa _ fl
+ ^12^2 + + "" + — 0
123 k such a system of equations can
^21^1 + ^22^2 + ^23^3 + "" + — 0
also be solved
i^i 1 ^21 ^3 1 1 ^fa _ t-j
+ ^12^2 + a13x3 + "" + —
a21xi1 + a22x*2 + a23x3:3 + "- + a2nx^n — ^ such a system of equations can
k^ 1 ^2 1 ^3 1 1 fcfa _ j
^31^1 + ^32^2 + Q33X3 + "" + Q3nXn — ^
also be solved
5. Conclusions
We considered a new way to calculate the values of the raskulants (non-square determinants).We independently introduced the definition of these concepts. A general formula was given for calculating the values of the raskulants. And then we considered the practical meanings of these formulas. With the help of raskulants, homogeneous and heterogeneous indefinite equations and systems of indefinite equations were solved. We have considered a new algorithm for finding the largest common divisor of several numbers.
REFERENCES
INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE "RAPID DEVELOPMENT OF MULTILATERAL INTERNATIONAL RELATIONS FROM A
PEDAGOGICAL POINT OF VIEW"
_4 JUNE, 2024_
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