Ratio Mathematica Volume 46, 2023

Super fuzzy matrix of inverse in kth Order

R.Deepa*
P. Sundararajan†

Abstract

In normal matrix and fuzzy matrix working well but sometimes fail
to work for classical model problems. Because, it is not satisfied the
consistency conditions and other parameter. Our proposed method to
satisfy the all condition including consistency and perform well com-
pared to the other existing models. Unexpected event modelling is a
affluent area of study in fuzzy matrix (FM) modelling. Every fuzzy
matrix may be shown as a multi-dimensional concept, but standard
matrices cannot achieve this without the proper scale. To solve this
issue, a certain kind of classical fuzzy matrix is required. In this study,
the idea of an inverse of a k-regular fuzzy matrix is introduced, and
some of its key characteristics are listed. As a result, the same regu-
larity indicator is used to describe a matrix. Investigation is also done
on the relationship between the regular, k-regular, and consistency of
fuzzy matrices powers.
Keywords: inverse; k-regular; super fuzzy matrix; Fuzzy matrix; reg-
ular
2020 AMS subject classifications: 03E72, 15A09, 35D10, 39B42,
58J52. 1

*Excel Engineering College(Autonomous), Komarapalayam, Tamilnadu, India. deepssengo
@gmail.com.

†Department of Mathematics, Arignar Anna Govt. Arts College, Namakkal. Tamilnadu, India.
ponsundar03@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1079. ISSN: 1592-7415. eISSN: 2282-8214. ©R.Deepa et al.
This paper is published under the CC-BY licence agreement.

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R.Deepa and P.Sundararajan

1 Introduction
Each component of a matrix known as a boolean matrix has a value of 0 or

1. Basic values for a fuzzy matrix fall between [0, 1]. By Kim and Roush, the
idea of fuzzy matrix sections was introduced. In all auothers contributed signifi-
cantly to fuzzy matrix augmentation. Later, utilising modified regular inverses of
the coefficient matrix, Zheng and Wang developed the widely used m x n fuzzy
linear machine and the inconsistent fuzzy linear system. Using matrix-modified
common inverse theory, Abbasbandy .S and . M [2005] investigated the minimal
response of the overall twin fuzzy linear device. We provide a unique way for find-
ing the inverse of a fantastic fuzzy matrix by changing a well-known idea in this
research D and H [1978],P [1963], Ravi.J and et. al. [2022], Kandasamy W.B.V.
and llanthenral K. [2007], M.Z. and E.G [1994], M.G [1977], K and B [2006], B
and K [2006].
The super fuzzy matrix’s initial and core notions are covered in Section 2 along
with an illustration. The method of our suggested notion is presented in Section
3. The main findings, theorems, and numerical examples of super fuzzy inverse
matrices of order k are presented in Section 4, and the conclusion is derived in
Section 5.

2 Preliminaries
noindent The core concepts and notations of a super fuzzy matrix will be de-

scribed in this section. This article concentrates on the same old number ordering,
underlying maxmin (min max) operations, and fuzzy matrices in super (SFM)
with [0,1] support. By (SF)mxn and (SF)n, respectively, all fuzzy matrices in
super of order m x n and n x n are represented. Space created by the row (or) col-
umn is indicated by the letters R(A) or C(A) (A) , Kaufmann and Gupta [1985],
J.B [1983], .

Definition 2.1. Deepa.R and Sundararajan.P [2020] A matrix containing entries
that fall between [0, 1] is referred to as a fuzzy matrix. In the case of matrices, we
have a fuzzy related different matrix.

Definition 2.2. Deepa.R and Sundararajan.P [2020] Let us deem C fuzzy matrix

C =
[
C11 C12 C
C21 C22 C23

]
The fuzzy submatrices C11 and C21 have the same number of columns because
they are fuzzy submatrices, along with C12, C13, C21, C22, and C23. The fuzzy
matrices C13 and C23, as well as the fuzzy submatrices C12 and C22, all have

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Super fuzzy matrix of inverse in kth order

equal columns. The second index of the fuzzy sub-matrices reveals this. The
number of rows in fuzzy submatrices C11, C12, and C13 is the same. There are
exactly the same number of rows in the C21, C22, and C23 fuzzy submatrices. as a
result, we currently possess C generic super fuzzy matrix.

C =




C11 C12 . . . C1n
C21 C22... C 2n

.

.

.
Cm1 Cm2... Cmn




where Cij’s i = 1, 2, . . . , m and j =1, 2, . . . , n.

Definition 2.3. The modified regular inverse of matrix solution of a positive set of
equations can be found by altering the regular inverse of a non-singular matrix.
The modified regular inverse of any (potentially square) matrix with complex com-
ponents may be found using any method. It is used here and in other programmes
to immediately solve linear matrix issues as well as to get an expression for the
principal idempotent components of the matrix.

3 Methodology
The kth regular super fuzzy matrix is examined in this section.

Definition 3.1. Let C =




c11 c12 . . . c1n
c21 c22... c 2n

.

.

.
cm1 cm2... cmn




It is a matrix of fuzzy that is Ck each aij is 0≤ aij ≤1. The exact strength of each
ingredient is examined here. It is the G-name. inverse’s.

Ck =




1−kc11 1−kc12 . . . 1−kc1n
1−kc21 1−kc22... 1−kc 2n

.

.

.
1−kcm1 1−kcm2... 1−kcmn




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R.Deepa and P.Sundararajan

Here, k is the smallest positive integer (Based on Probability). It is possible to
multiply two fuzzy matrices without requiring that they both be fuzzy. Every entry
that is positive is changed to a 1, and every entry that is negative is changed to a
0 or a tendency toward zero.

Definition 3.2. The Properties of Super fuzzy matrix is,(
Ak
)k

= A

AxAk = Ak,
(A+B)k= Ak + Bk,
(λA)k = λAk,
(BA)k= AkBk,
AAk=0 implies A=0.

Definition 3.3. The condition of the supper fuzzy matrix are given below,
AXA=A
XAX=X
(AX)k = AX
(XA)k = XA
XXkAk = X
XAAk=Ak

BAkAAk=Ak

X=XXkAk=XXkAkAY=XAY=XAAkYkY=AkYkY=Y.

4 Result & application
The Ck of super fuzzy matrix met all of the requirements for convergence.

Theorem 4.1. Let A be the inverse of the k by KRSFM (k regular super fuzzy
matrix), with non-zero rows representing the norm. If A meets the maximin con-
ditions for the matrix equation ASA=A for some super fuzzy matrix S, then C is k
regular.

Proof. The non-zero rows of the inverse of the KRSFM of C serve as the
standard basis in this case. If SB=Q, then Q’s rows are permutations of C’s rows.
Then, X is an idempotent of KRSFM, with the same row space as C and non-zero
Z rows functioning as a standard basis..
Since the standard foundation is different for each S, B=GQ.
Then
CST C=SQST SQ=SQQ=SQ=Ck

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Super fuzzy matrix of inverse in kth order

⇒CSC=Ck.
So, C is k regular.

Theorem 4.2. Let C,D (SF)nxm and k=p be two inverse of p regular super fuzzy
matrix. If C is p regular, then we prove that

Hence Di* =
∑

xijA
∗.

⇒D=XCp
⇒D=XCC’C (since CC’C=Cp)
⇒D=DC’Cp
⇒D=CC’C
⇒D=CC’CY.
⇒D=CpC’D.

Theorem 4.3. For C∈(SF)n and for any G−∈(SF )n, if CkX=CkG−, where X is a
{1K ,3K} inverse of C then, G− is a {1K ,3K} inverse of C.

Proof. Since X is a {1K ,3K}
⇒CkXC=Ck and (CkX)T =CkX.
The Post is multiplied by A on both sides of CkX=CkG− , CkG−C=CkXC=Ck.
(CkG−)T =(CkX)T =CkX=CkG− .
Hence G− is a {1K ,3K}inverse of C.

Theorem 4.4. For C∈(SF )n, X is a {1P ,3P} inverse of C and H− is a {1P ,3P}
inverse of C then, CpX=CpH−.

Proof. Since X is a {1P ,3P} inverse of C.
⇒CpXC=Cp and (CpX)T =CpX.
H− is a {1P ,3P} inverse of C,
⇒CH−Cp=Cp and (CH− )T =CH− .
CpH−=(CpXC)H−= (CpX)T (CH− )=(CpX)T (CH− )T =XT (CT )p(H− )T CT

= XT (CH− Cp)T =XT (Cp)T =(CpX)T =CpX.

Theorem 4.5. For D∈(SF)n, if DT D is D right KRSFM and R(Dp)⊆R(DT D)p then
D has a {1P ,3P} inverse. In particular for p=1, U=(DT D)−DT is a {1, 3} inverse
of D.

Proof. let (DT D)p(DT D)−(DT D) = (DT D)p
Since R(Dp)⊆R((DT D)p),
Dp=X(DT D)p for some X∈(SF)n Dnd tape U=(DT D)−DT
DpUD=(Dp)(UD)=(X(DT D)p)((DT D)−DT D)
=X((DT D)p(DT D)−(DT D))
= X(DT D)p=(Dp).
TDpe V=(DT D)−(Dp)T .

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R.Deepa and P.Sundararajan

DpV=(Dp)V
=(X(DT D)p)((DT D)−(Dp)T )
= X(DT D)p(DT D)−(DT D)pDT

=X(DT D)p(DT D)−(DT D)(DT D)p−1XT

= X(DT D)p(DT D)p−1XT

=X(DT D)2p−1XT =(X(DT D)2p−1XT )T = (DpV)T .
Hence D has D {1P ,3P}inverse.
In particular for p=1,Y= (DT D)−DT is D {1, 3} inverse of D.

Theorem 4.6. Let C∈(SF)n be a right KRSFM and R(CT C)k⊆R(Ck) then CT C
has a {3K}inverse.

Proof. Since, CkXC=Ck.
Since R((CT C)k)⊆R(Ck), (CT C)k=ZCk for some Z∈(SF )n and take Y=XC.
(CT C)kY=(ZCk)(XC)=Z(CkXC)=ZCk=(CT C)k=((CT C)k)T

=((CT C)kY)T

Hence CT C has a {3K} inverse.
The findings for the other associated fuzzy matrices improve, and the theorem

is fulfilled. While some techniques (regular and inverse) meet some conditions,
our proposed KRSFM meets all of them (regular & inverse).

Example 4.1. Let us consider IKSFM

A=


 0.5 0.7 0.60.3 0.2 0.2

0.4 0.3 0.1




The rows of A are separate and make up the foundation. IKSFPM is a form that

P =


 0 1 01 0 0

0 0 1




We know that APA=A. Now for IKSFM

R=


 0.1 0.4 0.20.3 0.4 0.2

0.3 0.1 0.1




There for RA=A. So g inverse of A sis

PR=


 0.3 0.4 0.20.1 0.4 0.2

0.3 0.1 0.1




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Super fuzzy matrix of inverse in kth order

Which is satisfy the relation AXA=A.

Example 4.2. Let A =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]
and

B =
[
< 0.6,0.2 > < 0.5,0.4 >
< 0.,7 0.1 > < 0.4,0.4 >

]
For which B=BAT A holds.

Example 4.3. Let IKSFM is,

C =


 < 0.8,0.2 > < 0.4,0.2 > < 0.3,0.2 >< 0.4,0.2 > < 0.4,0.2 > < 0.3,0.2 >
< 0.6,0.2 > < 0.6,0.2 > < 0.8,0.2 >




D =


 < 0.8,0.2 > < 0.4,0.3 > < 0.2,0.2 >< 0.4,0.3 > < 0.4,0.2 > < 0.2,0.2 >
< 0.4,0.2 > < 0.4,0.2 > < 0.6,0.2 >




E =


 < 0.7,0.2 > < 0.4,0.3 > < 0.3,0.3 >< 0.4,0.3 > < 0.4,0.2 > < 0.4,0.2 >
< 0.4,0.2 > < 0.6,0.2 > < 0.8,0.2 >




Be two of its g inverse of A.

Then X = CDE =


 < 0.8,0.2 > < 0.4,0.2 > < 0.3,0.2 >< 0.4,0.2 > < 0.4,0.2 > < 0.3,0.2 >
< 0.6,0.2 > < 0.6,0.2 > < 0.8,0.2 >




Forth above X, CXC=C and XCX=X holds. So X is a semi inverse of the IKSFM.

Example 4.4. Let us consider the symmetric IKSFM

A =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]
Now, A2 is,

A2 =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

] [
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]

=

[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]
= A
The above matrix is symmetric and idempotent.

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R.Deepa and P.Sundararajan

Example 4.5. Let us consider the symmetric IKSFM

A =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]
and

B =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0 > < 0,0.4 >

]
Now, A2 is,

B2 =
[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0 > < 0,0.4 >

] [
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0 > < 0,0.4 >

]

=

[
< 0.4,0.2 > < 0.4,0.5 >
< 0.8, 0.1 > < 0.3,0.4 >

]
6= B
The above matrix is not idempotent.

5 Conclusions
The main conclusions of the study may be presented in a short Conclusions

section, which may stand alone or form a subsection of a Discussion or Results.
In this paper, the inverse of the KRSFM is used to provide several innovative pro-
posals and theorems. The original machine is replaced with matrix coefficient A
by two distinct n m matrix equation systems. As a result, FM must resolve the is-
sue. The k-regular, the regularity of the fuzzy matrix powers, and the relationship
between each regular are all examples of regularities. Our approach addresses
any issues that need the inverse of k regular super fuzzy matrices. A appropriate
theorem is also shown. In the next essay, we will look at a variety of fundamen-
tal properties as well as computer vision applications. In our proposed classical
model to proved, it is a consistent. Also in future, we are trying this concept in
neural networking system and IOT.

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