Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 272 https://internationalpubls.com Secondary k-Kernel Symmetric Interval Valued Intuitionistic Neutrosophic Fuzzy Matrices M.Anandhkumar1, T. Harikrishnan2, S. M. Chithra3 M. John Peter4 V. Sathishkumar5 A. Bobin6 1Department of Mathematics, IFET College of Engineering (Autonomous), Villupuram, Tamilnadu, India. anandhkumarmm@gmail.com 2Department of Mathematics, Faculty of Science and Humanities, SRM Institute of Science and Technology, Ramapuram, Tamilnadu, India mokshihari2009@gmail.com 3Department of Mathematics, R.M.K College of Engineering and Technology, Chennai, Tamilnadu, India. chithra.sm@rmkcet.ac.in 4 Department of Mathematics, Panimalar Engineering College, Chennai – 600123, Tamil Nadu, India. johnpmath@gmail.com 5Department of Mathematics , Rajalakshmi institute of technology (Autonomous), Chennai; Tamilnadu, India vsathishkumar2020@gmail.com 6Department of Mathematics, IFET College of Engineering (Autonomous), Villupuram, Tamilnadu, India. bobinalbert@gmail.com Article History: Received: 30-07-2024 Revised: 20-09-2024 Accepted: 02-10-2024 Abstract We propose the idea of secondary symmetric k-kernels with interval-valued intuitionistic neutrosophic fuzzy matrices (IVINFM) like an EP matrix within the complex field. The notion of secondary kernel IVINFM and k-kernel symmetric (KS) IVINFM is presented by providing examples. We also illustrate the graphical representation of KS adjacency IVINFM and incidence IVINFM. We found every isomorphic IVINFM and non- isomorphic IVINFM graph to be k-KS IVINFM. However, the reverse shall not be the case. The definition of k-symmetric IVINFM as k- KS IVINFM, but the reverse is not the case. The characteristics of secondary kernels IVINFM, which are symmetric IVINFM, have been explored in this research using examples. The relationship between s-k KS IVINFM, symmetric s- KS IVINFM, and KS IVINFM, and the KS IVINFMs are examined. We identify the required and sufficient criteria for the KS IVINFM and s-k KS IVINFM. The comparable requirements for the various g-inverses that make up an s-k KS IVINFM are shown. The generalized inverses of a KS IVINFM A, which correspond to the sets A{1,2}, A{1, 2 ,3} and A{1,2, 4} are described. Keywords: IVINM, KS- IVINFM, s-k- KS IVINFM, adjacency IVINFM, incidence IVINFM. 1. Introduction Consider A as a neutrosophic matrix (NFM). If A is a part of (NF)n it is referred to as a k-KS NFM when N(A) = N(KATK). Matrices play a crucial role in various research fields within engineering and science. Conventional matrix theory must tackle problems involving significant uncertainty. Zadeh [1] introduced the concept of fuzzy sets (FS) using membership numbers. Atanassov [2] intuitively designed a fuzzy set that effectively provides membership and non-membership grades for an element. mailto:anandhkumarmm@gmail.com mailto:mokshihari2009@gmail.com mailto:johnpmath@gmail.com mailto:bobinalbert@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 273 https://internationalpubls.com Smarandache [3] proposed concepts like neutrosophic sets, a mathematical instrument to solve problems requiring indeterminacy, ambiguity, and inconsistent data. Fuzzy matrices (FM) can be used to address specific types of problems. Numerous researchers have completed many projects. Fuzzy matrices handle only membership values. They cannot deal with values that are non-members. Khan et.al [4] were the pioneers in introducing IFMs to the academic community. Anandhkumar et al. [20-21] investigated generalized symmetric NFMs, partial ordering concepts in NFMs. Atanassov [5,6] developed significant results related to IFS and IVIFS. Hashimoto [7] investigated the canonical type of transitive matrices. Pal and Susanta Kha [22] explored interval- valued intuitionistic properties. Vidhya and Irene Hepzibah [23] presented IVNFMs. Kim and Roush [8] explored generalized FM. Lee [9] focused on secondary skew-symmetric and secondary orthogonal matrices. Hill and Waters [10] addressed Hermitian matrices. Meenakshi [11] delved into the concept of fuzzy matrices and their properties. Meenakshi and Jaya Shree [12] explored symmetric k-kernel matrices, while Meenakshi and Krishanmoorthy [13] discussed the properties of secondary k-Hermitian matrices. Additionally, Meenakshi and Jaya Shree [14] have been investigating k-RS matrices. Morteza Yazdani et al. [24] applied interval-valued neutrosophic concepts to decision-making in supplier selection. Jaya Shree [15] delved into secondary K-KS fuzzy concepts, and Shyamal and Pal [16] presented findings on IVFM. Meenakshi and Kalliraja [17] provided a summary of regular interval-valued matrices. Anandhkumar and colleagues [18] conducted research on pseudo similarity neutrosophic fuzzy matrices.Anandhkumar et.ai[19] have studied on various Inverse of Neutrosophic Fuzzy Matrices. Jaya Shree [25] have discussed Secondary k-range symmetric fuzzy matrices. Vidhya and R. Irene Hepzibah [26] have focused On Interval Valued Neutrosophic Fuzzy Matrices. Anandhkumar et.al [27-33] have studied on Kernel and K-Kernel Symmetric Intuitionistic Fuzzy Matrices, IV Secondary k-RS Neutrosophic fuzzy matrices, secondary k-CS neutrosophic fuzzy matrices, Partial orderings, Characterizations and Generalization of k-idempotent Neutrosophic fuzzy matrices, Reverse Tilde (T) and Minus Partial Ordering on fuzzy matrices, Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices,Generalized Symmetric Fermatean Neutrosophic Fuzzy Matrices. We present the secondary K-KS IVINM and provide a few basic operators on IVINMs. Section 2 highlights on preliminaries. Section 3 on Graphical Representation of KS Adjacency IVINM is given. Section 4 discusses on s - k KS IVINMs and regular IVINMs. In Section 5, we have discussed on various generalized inverses of matrices in IVINM. The generalized inverses of a s−ks IVINFM equivalent to the sets A={1,2}, A={1,2,3}, A={1,2,4} are considered. Table: 1 Review of the literatures References Extension of Fuzzy Matrices. Year [12] On k- KSFM 2009 [14] On k -RSFM 2009 [15] Secondary κ- KSFM 2014 [25] Secondary κ-RSFM 2018 [28] IV Secondary k-RS Neutrosophic fuzzy matrices 2024 https://www.scopus.com/authid/detail.uri?authorId=58220200200 https://www.researchgate.net/scientific-contributions/D-Jaya-Shree-2140064332?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 274 https://internationalpubls.com [29] secondary k-CS neutrosophic fuzzy matrices 2024 [32] Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices 2024 Proposed Secondary k- KS IVINFMs 2024 Based on literature review that reflects no research has been carried out on Secondary k-KS IVINFM to overcome the research gap. 1.1 Research Gap Jayashri detailed on k-range symmetric fuzzy matrices (FMs). Meenakshi and Jayashri established the results of KS in FMs.Anandhkumar established the results of IV Secondary k-RS Neutrosophic fuzzy matrices. In this paper, we have used the idea of secondary k-KS IVINFM. We have implemented the properties in IVINFM. We first present similar characterizations of the Secondary k-KS IVINFM. We then define the case of a Secondary k-KS IVINFM. We have looked at various g-inverses that are associated with regular matrices. Then, we have the definition of a set of all inverses using secondary symmetric s-k-Kernel IVINFM. 1.2 Notations: [ , , ] T v LA A A  - Transpose of the IVINM[ , , ]v LA A A  , [ , , ] T v UA A A  - Transpose of the IVINM[ , , ]v UA A A  , [ , , ]v LA A A  + - Moore-Penrose inverse (MPI) of IVINM[ , , ]v LA A A  , [ , , ]v UA A A  + - MPI of IVINM[ , , ]v UA A A  , R ( )[ , , ]v LA A A  - Row space of [ , , ]v LA A A  R ( )[ , , ]v UA A A  - Row space of[ , , ]v UA A A  , C ( )[ , , ]v LA A A  -Column space of[ , , ]v LA A A  , C ( )[ , , ]v UA A A  -Column space of [ , , ]v UA A A  2. Preliminaries Consider V a permutation matrix with secondary diagonal units. The function κ(x)=(zk[1], zk[2], zk[3],…, zk[n])∈ Fn×1 for z = z1, z2,..,zn ∈F[1×n],, K is involuntary, The subsequent properties holds good by using the Definition of V [15]. (P.2.1) KK T = K T K = In, K = K T , K 2 = I (P.2.2) V =V T, VV T =V TV = In and V2 = I (P.2.3) N ( )[ , , ]v LA A A  = N ( )[ , , ]v LA A A  V, N ( )[ , , ]v LA A A  =N ( )[ , , ]v LA A A  K N ( )[ , , ]v UA A A  = N ( )[ , , ]v UA A A  V, N ( )[ , , ]v UA A A  =N ( )[ , , ]v UA A A  K Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 275 https://internationalpubls.com (P.2.4) N ( )[ , , ] V T v LA A A  = N ( )V[ , , ] T v LA A A  , N ( )V[ , , ] T v LA A A  =N ( )[ , , ] T v LA A A V  N ( )[ , , ] V T v UA A A  = N ( )V[ , , ] T v UA A A  , N ( )V[ , , ] T v UA A A  = ( )[ , , ] T v UA A A V  Definition:2.1[26] An IVNFM A = [xij,< aijµ, aij aijν >]m×n wherever aijµ, aij and aijν are the subsets of [0,1] which are represented by aijµ = [aijµL, aijµU], aij = [aijL, aijU] and aijν = [aijνL, aijνU] with the conditions 0≤aijµU + aijU+ aijνU ≤ 3, 0≤aijµL + aijL+ aijνL ≤ 3 , 0 ≤ aµL ≤ aµU ≤ 1, 0 ≤ aL ≤ aU ≤ 1, 0 ≤ aνL ≤ aνU ≤ 1 for i = 1,2,···,m and j = 1,2,···,n. Example2.1 Consider an IVNFM [1,1],[1,1],[0,0] [0.2,0.5],[0.3,0.6],[0.4,0.4] [0.2,0.5],[0.3,0.6],[0.4,0.4] [1,1],[1,1],[0,0] A      =        Lower Limit IVNFM, 1,1,0 0.2,0.3,0,4 [ , , ] , 0.2,0.3,0.4 1,1,0 v LA A A       =        Upper Limit IVNFM, 1,1,0 0.5,0.6,0.4 [ , , ] 0.5,0.6,0.4 1,1,0 v UA A A       =        [0,0],[1,1],[1,1] [0.2,0.4],[0.3,0.5],[0.1,0.5] and [0.2,0.4],[0.3,0.5],[0.1,0.5] [0,0],[1,1],[1,1] B      =        [1,1],[0,0],[1,1] [0.2,0.5],[0.3,0.5],[0.1,0.5] Then, [0.2,0.5],[0.3,0.5],[0.1,0.5] [1,1],[0,0],[1,1] A B      + =        Definition 2.2 [20] (Null IVNFM) IVNFM is said to be Null if the entries of true and indeterminacy are zero and the entries of false is one i.e., ([0,0],[0,0],[1,1]). Example: 2.2 Consider IVNFM      ( )      ( )      ( )      ( )      ( )      ( )      ( )      ( )      ( ) 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0 ,,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 0,0 , 0,0 , 1,1 A     =        Note:1 [11] For IVNFM A∈ Fn with det A > ([0,0], [0,0],[1,1]) where no rows or columns are zero, we have N(A) = ([0,0],[0,0],[1,1]) = N(AT). Additionally, for a symmetric matrix A = AT, it follows that N(A)= N(AT). Note:2 [12] Let A is k-Symmetric IVNFM implies it implies that it is also a k-KS IVNFM, such that A = K(AT)K which leads to N(A) = N(KAT K) .Example 2.3. Example 2.3 demonstrates that the converse is not necessarily true. Example: 2.3 Let us Consider IVNFM [0,1],[0,1],[0.5,0.5] [0,0.5],[0,0.6],[0.4,0.4] [0.3,0.5],[0.4,0.6],[0.5,0.5] [0.5,0.5],[0.4,0.6],[0.5,0.5] [0.1,0.3],[0.4,0.6],[0.6,0.6] [0,0.5],[0,0.6],[0.4,0.5] [0.4,0.5],[0.5,0.6],[0.3,0. A       =        5] [0.3,0.5],[0.4,0.4],[0.5,0.5] [0,0.5],[0,0.6],[0.3,0.5]              Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 276 https://internationalpubls.com (0,0,1) (0,0,1) (1,1,0) (0,0,1) (1,1,0) (0,0,1) (1,1,0) (0,0,1) (0,0,1) K     =      , (0,0,0.5) (0,0,0.4) (0.3,0.4,0.5) (0.5,0.4,0.5) (0.1,0.4,0.6) (0,0,0.4) (0.4,0.5,0.3) (0.3,0.4,0.5) (0,0,0.3) LA     =      Therefore, AL  KAL T K, But, N (AL) = N(KAL T K) = (0,0,1) Definition 2.3. [20] For IVNFM P is KS fuzzy matrix iff N ( )[ , , ]v LA A A  = N ( )[ , , ] T v LA A A  and N ( )[ , , ]v UA A A  = N ( )[ , , ] T v UA A A  . Definition 2.4. [27] An IVINFM A = [xij,< aijµ, aij aijν >]m×n where aijµ, aij and aijν are the subsets of [0,1] which are represented by aijµ = [aijµL, aijµU], aij = [aijL, aijU] and aijν = [aijνL, aijνU] with the conditions 0≤aijµU + aijU+ aijνU ≤ 2, 0 ≤ aijµU ≤ aijνU ≤ 1, 0 ≤aijU≤ 1 for i = 1,2,···,m and j = 1,2,···,n. 3. Graphical Representation of KS Adjacency IVINFM. Definition 3.1. Adjacency IVINFM: An IVINM is a square matrix representing a finite graph. The elements of this matrix indicate whether pairs of vertices in the graph are connected. In the case of a finite simple graph, the IVINFM can be represented as a binary matrix, typically denoted as ([1,1],[1,1],[0,0]) and ([0,0],[0,0],[1,1]), where the diagonal elements are consistently set to ([0,0],[0,0],[1,1]) .Let G(V, E) denote a graph with n vertices. The adjacency matrix A = [aij] is a symmetric matrix defined by i j ij ([1,1],[1,1],[0,0]) when v isadjacent to v [a ] ([0,0],[0,0],[1,1]) otherwise A  = =  , denoted by A(G) or AG Example: 3.1 Consider an IVINM and a equivalent graph 1 3 4 2 5 1 3 4 2 5 ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1 v v v v v v v v v v ,1],[0,0]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1])                    Definition 3.2. Incidence IVINFM Let G(V, E) represent a simple graph with n vertices. Let V = {V1, V2, …, Vn} and E = {e1, e2, ..., em}. Then, the incidence IVINFM I = [mij] is a n m matrix defined by i j ij ([1,1],[1,1],[0,0]) when v isincident toe [ ] ([0,0],[0,0],[1,1]) otherwise I m  = =  , denoted by A(G) or AG. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 277 https://internationalpubls.com Example:3.2 Consider an incidence IVINFM and a equivalent graph. The incidence IVINFM is Definition 3.3. Isomorphic of Graph Two graphs are said to be isomorphic if number of vertices, edges, degree sequence and adjacency IVINFM are equal. Relation between isomorphism, non-isomorphism and KS The adjacency IVINFM of the graph is given by ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1 a b c d e f a b c d e f ]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1])                       Consider the graph H and name as follows The adjacency IVINM of the graph is given by 1 2 3 4 5 6 1 ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) 2 ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1 3 4 5 6 ]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1])                       1 2 3 4 5 ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([0,0],[0,0], e e e e e a b A c d = [1,1]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0]) ([1,1],[1,1],[0,0])                Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 278 https://internationalpubls.com The two graphs presented have identical numbers of vertices, edges, and degree sequences, yet their adjacency IVINFMs differ. Therefore the given Graph is not isomorphic but KS. Let us form the adjacency IVINFM AG and AH 1 2 3 4 5 1 2 ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1] G v v v v v v v A = 3 4 ,[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0] v v 5 ) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1])v                    1 2 3 4 5 1 2 ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1] H u u u u u u u A = 3 4 ,[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0] u u 5 ) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1]) ([1,1],[1,1],[0,0]) ([0,0],[0,0],[1,1])u                    The two graphs provided have the same number of vertices, edges, and degree sequences, and their adjacency IVINFMs are also identical. Therefore, the graphs are isomorphic and also KS IVINFM. Every isomorphic and non-isomorphic graph is KS adjacency IVINM but converse need not be true. 4. Secondary k-KS IVINFM Definition 4.1 For an IVINM [ , , ] ,[ , , ]v L v U nnA IA A A A MA A IV N   =   is a s - symmetric IVINFM iff ( )[ , , ] [ , , ] T v L v LA A A V A A A V   = and [ , , ]v UA A A  ( )[ , , ] .T v UV A A A V = Definition 4.2 For an IVINM [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s- KS IVINFM iff ( ) ( )[ , , ] [ , , ] ,T v L v LN A A A N V A A A V   = ( )[ , , ]v UN A A A  = ( )[ , , ] .T v UN V A A A V  Definition 4.3. “For an IVINFM [ , , ] ,v LA A A A = [ , , ]v UA A A   is a s-k- KS IVINFM iff ( )[ , , ]v LN A A A  = ( )[ , , ] ,T v LN KV A A A VK  ( )[ , , ]v UN A A A  = ( )[ , , ] .T v UN KV A A A VK  Lemma 4.1. For an IVINM [ , , ] ,v LA A A A = [ , , ]v UA A A  IVNFMnn is a s- KS IVINM  VA [ , , ] ,v LV A A A = V[ , , ]v UA A A   KS -IVINFM  V [ , , ] Vv LA A A A = ,[ , , ] Vv UA A A   is a KS-IVINFM. Proof. Let [ , , ] ,[ , , ] IVNFMv L v U nnA A A A A A A   =  be a s-KS IVINFM.  ( ) ( )[ , , ] [ , , ] T v L v LN A A A N V A A A V   = [Definition 3.2] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 279 https://internationalpubls.com  ( ) ( )[ , , ] V [ , , ] T v L v LN A A A N A A A V   =  [ , , ] Vv LA A A  is KS. [By P.2.2]  ( ) ( )V[ , , ] VV [ , , ]T T v L v LN A A A N VV A A A V   =  ( ) ( )V[ , , ] V[ , , ] T v L v LN A A A N A A A   =  V[ , , ]v LA A A  is KS. Similarly  ( ) ( )[ , , ] [ , , ] T v U v UN A A A N V A A A V   =  ( ) ( )[ , , ] V [ , , ] T v U v UN A A A N A A A V   =  [ , , ] Vv UA A A  is KS.  ( ) ( )V[ , , ] VV [ , , ]T T v U v UN A A A N VV A A A V   =  ( ) ( )V[ , , ] V[ , , ] T v U v UN A A A N A A A   =  V[ , , ]v UA A A  is KS. Hence the theorem. Example 4.1 Let us consider IVINFM [1,1],[0,0],[0,0] [0.2,0.5],[0.3,0.6],[0.4,0.4] [0.2,0.5],[0.3,0.6],[0.4,0.4] [1,1],[0,0],[0,0] A      =        Lower Limit IVINM, 1,0,0 0.2,0.3,0,4 [ , , ] , 0.2,0.3,0.4 1,0,0 v LA A A       =        Upper Limit IVINM, 1,0,0 0.5,0.6,0.4 [ , , ] 0.5,0.6,0.4 1,0,0 v UA A A       =        V 0,0,0 1,1,0 , 1,1,0 0,0,0      =        K 1,1,0 0,0,0 , 0,0,0 1,1,0      =        1,1,0 0,0,0 0,0,0 1,1,0 1,0,0 0.2,0.3,0,4 0,0,0 1,1,0 1,1,0 0,0,0 0.2,0.3,0.4 1,0,0 0,0,0 1,1,0 1,1,0 0,0,0 1,1,0 0,0,0 0,0,0 1,1,0 T LKVA VK                  =                                                  T L LKVA VK A Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 280 https://internationalpubls.com Similarly, , T U UKVA VK A 1,1,0 0,0,0 1,0,0 0.2,0.3,0,4 1,1,0 0,0,0 0,0,0 1,1,0 0.2,0.3,0.4 1,0,0 0,0,0 1,1,0 LKA K                  =                        L LKA K A Similarly, U UA KA K N (AL) = ( ) 0,0,0T LN KVA VK =  The main point is that while A is symmetric under both IVINFM and KS IVINFM, it does not hold the more specific symmetries of κ-symmetric and s-κ-symmetric within the IVINFM framework. Example 4.2. Let us consider IVINFM, [0.2,0.6],[0.2,0.4],[0.3,0.6] [0.4,0.5],[0.3,0.3],[0.2,0.4] [0.4,0.5],[0.3,0.3],[0.2,0.4] [0.2,0.6],[0.2,0.4],[0.3,0.6] A      =        V 0,0,0 1,1,0 , 1,1,0 0,0,0      =        1,1,0 0,0,0 , 0,0,0 1,1,0 K      =        LLIVINM, 0.2,0.2,0.3 0.4,0.3,0.2 , 0.4,0.3,0.2 0.2,0.2,0.3 LA      =        ULIVINM, 0.6,0.4,0.6 0.5,0.3,0.4 0.5,0.3,0.4 0.6,0.4,0.6 UA      =        0.2,0.2,0.3 0.4,0.3,0.2 0.4,0.3,0.2 0.2,0.2,0.3 T L LKVA VK A      = =       A is symmetric, KS, s-κ-symmetric IVINFM and hence s- k- KS IVINFM. Theorem 4.1. Below statements are equal for IVINMnA (i) [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is s −  KS IVINFM. (ii) KV [ , , ] ,KV[ , , ]v L v UA KV A A A A A A   =  is KS IVINFM. (iii) KV [ , , ] ,[ , , ]v L v UA A A A KV A A A KV   =  is KS IVINFM. (iv) V [ , , ] ,V[ , , ]v L v UA V A A A A A A   =  is k- KS IVINFM. (v) K [ , , ] K,[ , , ] Kv L v UA A A A A A A   =  is s- KS IVINFM. (vi) AT is a s-k KS IVINFM. (vii) ( )N([ , , ] ) [ , , ] VK , N([ , , ] )T v L v L v UA A A N A A A A A A     = ( )[ , , ] VKT v UN A A A = (viii) ( )N([ , , ] ) [ , , ] VK , N([ , , ] )T T v L v L v UA A A N A A A A A A     = ( )R[ , , ] VKv UA A A = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 281 https://internationalpubls.com (ix) ( )N(KV[ , , ] ) KV[ , , ] , N(KV[ , , ] ) T T v L v L v UA A A N A A A A A A     = ( )KV[ , , ] T T v UN A A A = (x) ( ) ( )VK KV[ , , ] VK, KV[ , , ] VK T T T T v L v UA N A A A N A A A   = is KS IVINFM. (xi) V [ , , ] V,[ , , ] Vv L v UA A A A A A A   =  is KS IVINFM. (xii) VKP VK[ , , ] ,VK[ , , ]v L v UA A A A A A   =  is KS IVINFM. (xiii) KA K[ , , ] ,K[ , , ]v L v UA A A A A A   =  is IV KS IVINFM. Proof: (i) iff (ii) iff (iv) Let [ , , ] ,[ , , ] IVNFMv L v U nnA A A A A A A   =  is s −  KS IVINFM Let [ , , ]v LA A A  is a s −  KS IVINFM. ([ , , ] ) ( [ , , ] ), N([ , , ] ) ( [ , , ] ),T T v L v L v U v UN A A A N KV A A A VK A A A N KV A A A VK        = = (By Definition 3.3) ( [ , , ] ) ( [ , , ] ) , N([ , , ] ) ( [ , , ] )T T v L v L v U v UN KV A A A N KV A A A A A A N KV A A A        = = By (P.2.3)  KV [ , , ] ,KV[ , , ]v L v UA KV A A A A A A   =  is KS IVINFM VP [ , , ] ,V[ , , ]v L v UV A A A A A A    =  is - KS IVINFM Therefore, (i), (ii), and (iv) are all equivalent. (i) iff (ii) iff (v) Let [ , , ] ,[ , , ] IVNFMv L v U nnA A A A A A A   =  is s −  KS IVINFM ( [ , , ] ) ( [ , , ] ) , N( [ , , ] ) ( [ , , ] ) ,T T v L v L v U v UN KV A A A N KV A A A KV A A A N KV A A A        = = ( ( [ , , ] )) (( )[ , , ] ( ) )T T v L v LN VK KV A A A N VK A A A VK VK    = , N( ( [ , , ] )) (( )[ , , ] ( ) )T T v U v UVK KV A A A N VK A A A VK VK   = [ , , ] ,[ , , ]v L v UAKV A A A KV A A A KV      =   is KS IVINFM [ , , ] ,[ , , ]v L v UPK A A A K A A A K      =   is s- KS IVINFM Therefore, (i), (iii), and (v) are all equivalent. (ii) (ix) [ , , ] , [ , , ]v L v UKVA KV A A A KV A A A     =   is KS IVINFM Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 282 https://internationalpubls.com ( ) ( )( ) ( )[ , , ] [ , , ] , N [ , , ] T v L v L v UN KV A A A N KV A A A KV A A A      = ( )( )[ , , ] T v UN KV A A A = (ii)  (ix) is true. (ii)  (vii) [ , , ] , [ , , ]v L v UKVA KV A A A KV A A A     =   is KS IVINFM ( ) ( )( ) ( )[ , , ] [ , , ] , [ , , ] T v L v L v UN KV A A A N KV A A A N KV A A A      = ( )( )[ , , ] T v UN KV A A A = ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK A A A N A A A VK        = = Therefore, (ii), and (vii) are equivalent. . (iii)  (viii) [ , , ] ,[ , , ]v L v UAVK A A A VK A A A VK     =   ( ) ( )( ) ( )[ , , ] [ , , ] , N [ , , ] T v L v L v UN A A A VK N A A A VK A A A VK      = ( )( )[ , , ] T v UN A A A VK = ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ] T T v L v L v U v UN A A A VK N A A A A A A VK N A A A        = = Therefore, (iii), and (viii) are equivalent. (i)  (vi) Let [ , , ] ,[ , , ] IVNFMv L v U nnP A A A A A A   =  is a s −  KS IVINFM ([ , , ] ) ( [ , , ] ), N([ , , ] ) ( [ , , ] ),T T v L v L v U v UN A A A N KV A A A VK A A A N KV A A A VK        = = (By Definition 3.3)  ( ) ( [ , , ] , [ , , ] )T T v L v UKVA KV A A A KV A A A   = is KS IVINFM  ([ , , ] ,[ , , ] )T v L v UA VK A A A VK A A A VK   = is KS IVINFM  ( )[ , , ] ,[ , , ]T T T v L v UA A A A A A A   = is s −  KS IVINFM Therefore, (i), and (vi) are equivalent. Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s −  KS IVINFM. Consider [ , , ]v LA A A  is a s −  KS IVINFM ([ , , ] ) ( [ , , ] ), ([ , , ] ) ( [ , , ] ),T T v L v L v U v UN A A A N KV A A A VK N A A A N KV A A A VK        = = (By Definition 3.3) ([ , , ] ) ([ , , ] ), ([ , , ] ) ([ , , ] )v L v L v U v UN A A A VK N A A A VK N A A A VK N A A A VK        = = By (P.2.3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 283 https://internationalpubls.com [ , , ] ,[ , , ]v L v LAVK A A A VK A A A VK      =   is KS IVINFM [ , , ] ,[ , , ]v L v UAV A A A V A A A V      =   is -KS IVINFM Therefore, (i)  (x)  (xi) is true. (i)  (xii)  (xiii) Let [ , , ] ,[ , , ] IVNFMv L v U nnP A A A A A A   =  is a s −  KS IVINFM ([ , , ] ) ( [ , , ] ), ([ , , ] ) ( [ , , ] ),T T v L v L v U v UN A A A N KV A A A VK N A A A N KV A A A VK        = = [By Definition 3.3] ( [ , , ] ) ( [ , , ] ) , ( [ , , ] ) ( [ , , ] ) ,T T v L v L v U v UN VK A A A N VK A A A N VK A A A N VK A A A        = = By (P.2.3) ( ( [ , , ] )) (( )[ , , ] ( ) )T T v L v LN KV VK A A A N KV A A A KV KV    = , ( ( [ , , ] )) (( )[ , , ] ( ) )T T v U v UN KV VK A A A N KV A A A KV KV   = ( [ , , ] ) ( [ , , ] ) , ( [ , , ] ) ( [ , , ] )T T v L v L v U v UN VK A A A N VK A A A N VK A A A N VK A A A        = = [By Lemma. 2.2] [ , , ], [ , , ]v v UVKA VK A A A VK A A A      =   is a KS IVINM [ , , ] , [ , , ]v L v UKA K A A A K A A A      =   is a KS s- KS IVINM Therefore, (i), (xii),(xiii) are equivalent. Corollary:4.1 Below conditions are equal for IVINMnnA (i) [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is s-KS IVINFM. (ii) [ , , ] ,V[ , , ]v L v UVA V A A A A A A   =  is KS IVINFM. (iii) [ , , ] V,[ , , ] Vv L v UAV A A A A A A   =  is KS IVINFM. (iv) [ , , ] ,[ , , ]T T T v L v UA A A A A A A   =  is s − KS IVINFM. (v) ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A V A A A N A A A V       = = (vi) ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ] VT T v L v L v U v UN A A A N A A A V A A A N A A A       = = (vii) ( ) ( ) ( ) ( )KV[ , , ] V[ , , ] , N KV[ , , ] V[ , , ] T T v L v L v U v UN A A A N A A A A A A N A A A       = = Proof: (i) and (ii) implies (iii) Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s −  KS IVINFM ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK N A A A N A A A VK        = = ( ) ( )[ , , ] [ , , ] ,T v L v LN K A A A K N K A A A K    = [By Theorem 3.1] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 284 https://internationalpubls.com ( ) ( )[ , , ] [ , , ] T v U v UN K A A A K N K A A A K   = (i) & (ii)  (iii) is correct (i)& (iii)  (ii) [ , , ] ,[ , , ]v L v UA A A A A A A   =  is a -KS IVINFM ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN A A A N K A A A K N A A A N K A A A K        = = ( ) ( )( ) ( ) ( )( )[ , , ] [ , , ] , [ , , ] [ , , ] T T v L v L v U v UN K A A A K N A A A N K A A A K N A A A        = = Therefore, (i) & (iii) ( ) ( )( ) ( )[ , , ] [ , , ] , [ , , ] T v L v L v UN K A A A K N V A A A K N K A A A K      = ( )( )[ , , ] T v UN V A A A K = ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK N A A A N A A A VK        = = ( ) ( )( ) ( ) ( )( )[ , , ] [ , , ] , [ , , ] [ , , ] T T v L v L v U v UN A A A N KV A A A N A A A N KV A A A        = = [ , , ] ,[ , , ] IVIN  is a s-k-KS IVINM        M   v L v U nnA A A A A A A   =   (ii) is true (ii) & (iii) implies (i) is a s-k-KS  IVINM[ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK N A A A N A A A VK        = = ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN K A A A K N K A A A K N K A A A K N K A A A K        = = Therefore, (ii) and ( iii) ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN K A A A K N A A A N K A A A K N A A A        = = ( ) ( ) ( ) ( )[ , , ] [ , , ] , [ , , ] [ , , ]T T v L v L v U v UN A A A N K A A A K N A A A N K A A A K        = = [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a -KS IVINFM. Therefore, (i) is true, hence the theorem is proved. Theorem 4.2. For [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  Then, any two of the following conditions imply the third. (i) [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a - KS IVINFM. (ii) [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s-- KS IVINFM. (iii) ( ) ( ) ( )[ , , ] VK[ , , ] , N [ , , ] T T T v L v L v UN A A A N A A A A A A     = ( )VK[ , , ] T v UN A A A = Proof: (i) and (ii) implies (iii) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 285 https://internationalpubls.com Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s −  KS IVINFM ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK A A A N A A A VK        = = [By Theorem 3.1] ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ]T v L v L v UN K A A A K N K A A A K K A A A K      = ( )[ , , ] T v UN K A A A K = ( ) ( )( )N [ , , ] [ , , ] T T v U v UA A A N VK A A A   = (i) & (ii)  (iii) is correct (i)& (iii)  (ii) [ , , ] ,[ , , ]v L v UA A A A A A A   =  is an IV - KS ( ) ( )( ) ( ) ( )( )[ , , ] [ , , ] , N [ , , ] [ , , ] T T v L v L v U v UN K A A A K N A A A K A A A K N A A A        = = Therefore, (i) & (iii) ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN A A A N A A A VK A A A N A A A VK        = = ( ) ( )( ) ( ) ( )( )[ , , ] [ , , ] , N [ , , ] [ , , ] T T v L v L v U v UN A A A N KV A A A A A A N KV A A A        = = is a s-k-KS IV M[ , , ] ,[ , , ] IVIN IM Nv L v U nnA A A A A A A   =   (ii) is correct (ii) & (iii) (i) [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a s- - KS IVINM ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN K A A A K N K A A A K K A A A K N K A A A K        = = Therefore,(ii) and ( iii) ( ) ( ) ( ) ( )[ , , ] [ , , ] , N [ , , ] [ , , ]T T v L v L v U v UN A A A N K A A A K A A A N K A A A K        = = [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is a - KS IVINFM. Consequently, (i) is true, Therefore the theorem is proved. 5. s −  KS regular IVINFM In this segment, we have explored different generalized inverses of matrices within the IVINFM framework. We have also defined the criteria for various g-inverses of an s-k KS IVINFM to qualify Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 286 https://internationalpubls.com as s-k KS IVINFM. The generalized inverses of an s-κ KS IVINFM A equivalent to the sets A{1, 2}, A{1, 2, 3}, and A{1, 2, 4} have been specifically considered. Theorem 5.1: Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  , Z belongs to A{1,2} and AZ, ZA are a s- κ- KS IVINM. Then A is a s- κ - KS IVINM iff [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s- κ – KS IVINFM. Proof: Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  ( ) ( ) ( )[ , , ] [ , , ] Z[ , , ] Z[ , , ]v L v L v v LN A A A A A A A A A A A AKV N KV N       =  ( ) ( ) ( )[ , , ] [ , , ] [ , , ]v L v L v LA A A N AN ZVV ZVKKV N KA A AV A A     =   ( ) ( )[H ,en ,ce, , ] Z[  , ]N v L v LA A A A AV N AK    = ( )( )    N        ZA is s- κ-KS IVINMZ[ , , ] T v LKV KA A VA = ( )[ , , ] [Z , Z , Z ] N v L v T L T VKA A A   = ( )[Z , Z , ZN ] T v L VK = ( )( )[Z , Z , N Z ]v L T KV  = ( )( ) ( )[ , , ] [ , , ] T v v L T L KN AKV NA A VA A A   = ( )    N         VP is s- κ – SIVINM[ , , ] [Z , Z , Z ]v L v LK AV A KA   = ( )[Z , Z , N Z ]v LKV  = Similarly, ( ) ( )( ) ( )Hence, N KVZ is a S IVINM[Z , Z , Z ] [ , , ] T v U v UKV N KV A KA A   = ( ) ( )( )[ ,[ , , ] , , ] T v L v LN KV NA KVA A A A A    = ( ) ( )( )[ , , ] [ , , ] T v U v UKV NN KA VA A A A A   = ( ) ( )( )[Z , Z , Z ] [Z , Z , Z ] , T v L v LN KV N KV    = ( ) ( )( )[Z , Z , Z ] [Z , Z , Z ] T v U v UKV N KN V   = ,[Z , Z , Z ] [Z , Z , Z ]v L v UKVX KV KV      =   is a KS IVINFM [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a KS IVINFM. Theorem 5.2:, Let [ , , ] ,[ , , ] ,v L v UA A A A A A A   =  [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  ∈A{1,2,3}, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 287 https://internationalpubls.com ( )N(KV[ , , ] ) KV[X ,X ,X ] T v L v LA A A N   = , N(KV[P , P , P ] )v U  = ( )KV[Z , Z , Z ] T U U vUN   .Then [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  is s-κ- KS IVINFM  [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is s- κ – KS IVINFM. Proof: Given A{1,2,3}, Hence , ,[ , , ] [Z , Z , Z ] [ , , ] [ , , ]v L v L v L v LA A A A A A A A A       = [Z , Z , Z ] [ , , ] [Z , Z , Z ] [Z , Z ] ,, Zv L v L v L v LA A A       = ( )[ , , ] [Z , Z , Z ] [ , , ] [Z , Z , Z ]v L v L v L v L T A A A A A A       = ( )( ) ( ) [ , , ] [Z , Z , Z AC ] [onsider, N N   By using A, AZ, ] T v L v L v L T TK A A A A KA AV V     = = ( )( )[ , , ] [Z , Z , Z N ]  T v L v LA A AKV    = ( )( )  2.3 N     [ ,                   , ] [Z , Z ., Z ]v L v L T A A By PA   = ( )  N   [ , , ] [Z , Z , Z ]v L v LA A A   = ( )[Z , Z , N   Z ]v L = [Z , Z , Z ] [Z , Z , Z ] [ , ,By usi ] [Z , Z ,g ] Zn v L v L v L v LA A A       =   ( )  2.3 N                      [Z , Z , Z ]      v LKV By P = ( )( ) ( )Similarly, we can consider, N N  [ , , ] [Z , Z , Z ] [ , , ]U T T v U v v U TA A A A AKV VKA     = ( )( )[ , , ] [ Z Z ]N Z , , T v U v UA AK AV    = ( )( )  2.3 N     [ ,                   , ] [Z , Z ., Z ]v U v U T A A By PA   = ( ) ( ) N                           [ , , ] [Z , Z Z, Z P] T v U v UA A A PZ        = = ( )    N                               By using Z [Z , Z , AZ ] Zv U Z = = ( )  2.3  N                       [Z , Z ,    Z ] By v UKV P = If KVA is a KS IVINM ( ) ( )( )N N  ,[ , , ] [ , , ] T v L v LKV VA A A A AK A    = ( ) ( )( )N N ,   [ , , ] [ , ] T v U v UKV KVA A A A A A   = ( ) ( )( )N [Z , Z , Z ] [Z ] ,Z ,  ,N Z T v L v LKV KV    = [Z , Z , Z ] [Z , Z , Z[ ], ] v L v UKVX KV KV   = is a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 288 https://internationalpubls.com KS IVINM. [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s-k KS IVINM. Theorem 5.3: Let [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  , Z ∈ A {1, 2, 4}, (KV[ , , ] )N T v LA A A  ( ) ( )KV[Z , Z , Z ] , (KV[ , , ] ) KV[ZN , Z , ZN ]N T v L v U v UA A A     = = . Then KVP is an s- κ-KS IVINM iff [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s- κ- KS IVINFM. Proof: Given, A {1, 2, 4}, Hence ,[ , , ] [Z , Z , Z ] [ , , ] [ , , ]v L v L v L v LA A A A A A A A A       = ,[Z , Z , Z ] [ , , ] [Z , Z , Z ] [Z , Z , Z ]v L v L v L v LA A A       = ( )[Z , Z , Z ] [ , , ] [Z , Z , Z ] [ , , ]v L v L v L v L T A A A A A A       = ( )( ) ( )  [ , , ] [Z , Z , Z ] AC [ ,onsider, N N       By using A A] , Zv L v L T v L T TV A A A A A VKAK      = = ( )( )[ , , ] [Z , , Z ]N Z T v L v LKV A A A   = ( )( )  2.3[ , , ] [Z , Z , Z ]N v L T v LA A A By P   = ( )[ , , ] [Z , Z , ZN ]v L v LA A A   = ( )[Z , Z ,N Z ]v L = ( )  2.3KV[Z , Z ,N Z ]v L By P = ( )( ) ( )  N N       By using A AZA[ , , ] [Z , Z , Z ] [ , , ] T v U v U v T U TKV VA A A A A KA     = = ( )( )[ , , ] [Z , Z , Z ]N   v U v U T KV A A A   = ( )( )  2.3[ , , ] [Z , Z , Z ]N v U T v UA A A By P   = ( )[ , , ] [Z , Z , Z ] (PN Z)T v U v UA A A PZ     = =  ( )[X ,X ,X ]U U vUN  = ( )  2.3KV[Z , Z ,N Z ]v U By P = If KVP is a KS IVINM ( ) ( )( )N N  ,[ , , ] [ , , ] T v L v LKV VA A A A AK A    = ( ) ( )( )N N ,   [ , , ] [ , ] T v U v UKV KVA A A A A A   = ( ) ( )( ) N[Z , Z , Z ] [ ,Z , Z , ]  Z T v L v LN KV KV    = [Z , Z , Z ] [Z , Z , Z[ ], ] v L v UKVX KV KV   = is a KS IVINM. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 289 https://internationalpubls.com [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s-k KS IVINM. Corollary 5.1: For [ , , ] ,[ , , ] IVNFMv L v nnA A A A A A A   =  , Z A {1, 2} and [ , , ] [Z , Z , Z ]v L L L vLA A A AZ    = ,[ , , ] [Z , Z , Z ] ,v U v UA A A     [Z , Z , Z ] [ , , ] ,[Z , Z , Z ] [ , , ] ,v L v L v U v UZ A A A AA A A       =  is a s- KS IVINFM. Then A is a s- KS IVINM iff [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s- KS IVINFM. Corollary 5.2: For [ , , ] ,[ , , ] IVINM ,v L v U nnA A A A A A A   =  Z A {1, 2, 3}, N(KV[ , , ] )v LA A A  ( )V[Z , Z , Z ] , N(KV[ , , ]) T v L vN A A A   = ( )V[Z , Z , Z ] . T v UN  = Then A is a s- RS IVINM iff [Z , Z , Z ] ,[Z , Z , Z ]v L v UZ    =  is a s- KS IVINFM. Corollary 5.3: For [ , , ] ,[ , , ] IVINMv L v U nnA A A A A A A   =  , Z A {1, 2, 4} , N(V[ , , ] )T v LA A A  ( )V[Z , Z , Z ] , N(V[ , , ] ) T T v L v UN A A A   = ( )V[Z , Z , Z ] .v UN  = Then A is a s- KS IVINM iff Z is a s-KS IVINFM. 5. Conclusion: We present equivalent definitions of a k-KS IVINFM, KS IVINFM, and the s-KS variant of IVINFM and s-k KS IVINFM. We also give an example of s-k-symmetric IVINFM and s-k KS IVINFM, but the converse may not be valid. We review various g-inverses related to regular matrices and characterize the set of all inverted inverses. Statements for different G-inverses of a s-k-KS IVINFM and s-kernel IVINFM that is symmetric are identified. In the future, we will determine some properties related to secondary IVINFMs with k-KS. Refrences [1]. Zadeh L.A., Fuzzy Sets, Information and control.,(1965),,8, pp. 338-353. [2]. K.Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag, 1999. [3]. Smarandache,F, Neutrosophic set, a generalization of the intuitionistic fuzzy set. Int J Pure Appl Math.; .,(2005),.24(3):287–297. [4]. M.Pal, S.K.Khan and A.K.Shyamal, Intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets, 8(2) (2002) 51- 62. [5]. K.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986) 87-96. [6]. K.Atanassov, Operations over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 64 (1994) 159-174. [7]. H.Hashimoto, Canonical form of a transitive matrix, Fuzzy Sets and Systems, 11 (1983),157-162. [8]. K.H.Kim and F.W.Roush, Generalised fuzzy matrices, Fuzzy Sets and Systems, 4 (1980) 293315. [9]. A.Lee, Secondary Symmetric, Secondary Skew Symmetric, Secondary Orthogonal Matrices, Period Math, Hungary, 7 (1976) 63-76. [10]. R.D. Hill and S.R.Waters, On k-Real and k-Hermitian matrices, Linear Algebra and its Applications, 169 (1992) 17-29. [11]. AR.Meenakshi, Fuzzy Matrix: Theory and Applications, MJP Publishers, Chennai, 2008. [12]. AR.Meenakshi and D.Jaya Shree, On k-KS matrices, International Journal of Mathematics and Mathematical Sciences, 2009, Article ID 926217, 8 Pages. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 290 https://internationalpubls.com [13]. AR.Meenakshi and S.Krishanmoorthy, On Secondary k-Hermitian matrices, Journal of Modern Science, 1 (2009) 70-78. [14]. AR.Meenakshi and D.Jaya Shree, On K -range symmetric matrices, Proceedings of the National conference on Algebra and Graph Theory, MS University, (2009), 58- 67. [15]. D.Jaya shree , Secondary κ-KS Fuzzy Matrices, Intern. J. Fuzzy Mathematical Archive Vol. 5, No. 2, 2014, 89-94 ISSN: 2320 –3242 (P), 2320 –3250 ,Published on 20 December 2014. [16]. A. K. Shyamal and M. Pal, Interval valued Fuzzy matrices, Journal of Fuzzy Mathematics 14(3) (2006), 582-592. [17]. A. R. Meenakshi and M. Kalliraja, Regular Interval valued Fuzzy matrices, Advance in Fuzzy Mathematics 5(1) (2010), 7-15. [18]. Anandhkumar, M., Kamalakannan, V., Chithra, S.M., Said, B., “Pseudo Similarity of Neutrosophic Fuzzy matrices”, International Journal of Neutrosophic Science, Vol. 20, No. 04, PP. 191-196, 2023. [19]. Anandhkumar, M., Kanimozhi, B., Chithra, S.M., Kamalakannan, V., Said, B., “On various Inverse of Neutrosophic Fuzzy Matrices”, International Journal of Neutrosophic Science, Vol. 21, No. 02, PP. 20-31, 2023. [20]. M.Anandhkumar; G.Punithavalli; T.Soupramanien; Said Broumi, Generalized Symmetric Neutrosophic Fuzzy Matrices, Neutrosophic Sets and Systems, Vol. 57,2023, 57, pp. 114–127. [21]. Anandhkumar, M., Harikrishnan, T., Chithra, S.M., ...Kanimozhi, B., Said, B. “Reverse Sharp and Left-T Right-T Partial Ordering on Neutrosophic Fuzzy Matrices”International Journal of Neutrosophic Science, 2023, 21(4), pp. 135–145. [22]. M Pal and Susanta K. Khan Interval-Valued Intuitionistic Fuzzy Matrices, NIFS 11 (2005), 1, 16-27. [23]. R. Vidhya and R. Irene Hepzibah On Interval Valued Neutrosophic Fuzzy Matrices, Advances and Applications in Mathematical Sciences Volume 20, Issue 4, February 2021, Pages 561-57. [24]. Morteza Yazdani , Ali EbadiTorkayesh , Željko Stević , Prasenjit Chatterjee d, Sahand Asgharieh Ahari b, Violeta Doval Hernandez,An interval valued neutrosophic decision-making structure for sustainable supplier selection,Expert Systems with Applications, Volume 183, 30 November 2021. [25]. D. Jaya Shree, Secondary κ-range symmetric fuzzy matrices, Journal of Discrete Mathematical Sciences and Cryptography 21(1):1-11,2018. [26]. R. Vidhya and R. Irene Hepzibah, On Interval Valued Neutrosophic Fuzzy Matrices Advances and Applications in Mathematical Sciences Volume 20, Issue 4, February 2021, Pages 561-575. [27]. G. Punithavalli, M. Anandhkumar, Kernel and K-Kernel Symmetric Intuitionistic Fuzzy Matrices,TWMS J. App. and Eng. Math. V.14, N.3, 2024, pp. 1231-1240. [28]. M. Anandhkumar, G. Punithavalli, R. Jegan, and S Broumi, IV Secondary k-RS Neutrosophic fuzzy matrices, Neutrosophic Sets and Systems 61, 2024,1. [29]. M. Anandhkumar, G. Punithavalli, and E.Janaki, secondary k-CS neutrosophic fuzzy matrices, Neutrosophic Sets and Systems 64, 2024, pp. 24-37. [30]. M. Anandhkumar, T. Harikrishnan, S. M. Chithra, V. Kamalakannan, B. Kanimozhi, Partial orderings, Characterizations and Generalization of k-idempotent Neutrosophic fuzzy matrices, International Journal of Neutrosophic Science, Vol. 23, no. 2, 2024, pp. 286-295. [31]. M.Anandhkumar, B.Kanimozhi, S.M. Chithra, V.Kamalakannan, Reverse Tilde (T) and Minus Partial Ordering on Intuitionistic fuzzy matrices, Mathematical Modelling of Engineering Problems, 2023, 10(4), pp. 1427–1432. [32]. M. Anandhkumar, H. Prathab, S. M. Chithra, A. S. Prakaash, A. Bobin, Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices, International Journal of Neutrosophic Science, vol. 23, no. 4, 2024, pp. 23-28. [33]. Anandhkumar, M.; A. Bobin; S. M. Chithra; and V. Kamalakannan. "Generalized Symmetric Fermatean Neutrosophic Fuzzy Matrices." Neutrosophic Sets and Systems 70, 1 (2024). https://www.scopus.com/authid/detail.uri?authorId=58220200200 https://www.scopus.com/authid/detail.uri?authorId=58220200300 https://www.scopus.com/authid/detail.uri?authorId=57215184099 https://www.scopus.com/authid/detail.uri?authorId=55919041000 https://www.scopus.com/authid/detail.uri?authorId=58220200200 https://www.scopus.com/authid/detail.uri?authorId=58605820200 https://www.scopus.com/authid/detail.uri?authorId=57215184099 https://www.scopus.com/authid/detail.uri?authorId=58220200300 https://www.scopus.com/authid/detail.uri?authorId=55919041000 https://www.scopus.com/authid/detail.uri?authorId=58220200200 https://www.scopus.com/authid/detail.uri?authorId=58602942600 https://www.scopus.com/authid/detail.uri?authorId=57215184099 https://www.scopus.com/authid/detail.uri?authorId=58605820200 https://www.scopus.com/authid/detail.uri?authorId=55919041000 https://www.sciencedirect.com/journal/expert-systems-with-applications https://www.sciencedirect.com/journal/expert-systems-with-applications/vol/183/suppl/C https://www.researchgate.net/scientific-contributions/D-Jaya-Shree-2140064332?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19 https://www.researchgate.net/journal/Journal-of-Discrete-Mathematical-Sciences-and-Cryptography-2169-0065?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19 https://www.researchgate.net/journal/Journal-of-Discrete-Mathematical-Sciences-and-Cryptography-2169-0065?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19