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We explore the determinant theory for Quadri-Partitioned Neutrosophic Fuzzy Matrices (QPNFMs), investigating their properties. In this study, we establish that det (Padj (P)) = det (P) = det (adj (P)P). Additionally, we propose a refined method to compute the determinant of matrices with a higher number of rows and columns. Furthermore, an algorithm is developed to address decision-making problems based on QPNFMs. An illustrative example is provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR] more...

The objective of this study is to establish the results concerning Interval-Valued (IV) Secondary k-Range Symmetric (RS) Quadri Partitioned Neutrosophic Fuzzy Matrices (QPNFM). We have applied the RS condition within the neutrosophic environment to explore the relationships between IVQP s-k-RS, s-RS, IVQP k-RS, and IVQP RS matrices. This analysis has yielded significant insights into how these various matrix types interrelate and their structural properties. We have established the necessary and sufficient criteria for IVQP s-k-RS IVQPNFM, along with various generalized inverses of an IVQP s-ks-RS fuzzy matrix to maintain its classification as an IVQP s-k-RS matrix. Furthermore, we have characterized the generalized inverses of an IVQP s-k-RS matrix S corresponding to the sets S = {1,2}, S = {1,2,3}, and S = {1, 2, 4}. This characterization contributes to the foundational understanding of generalized inverses in the context of IVQPNFM. Additionally, a graphical representation of RS, Column symmetric (CS), and kernel symmetric (KS) adjacency and incidence QPNFM is illustrated. It is shown that every adjacency QPNFM is symmetric, RS, CS, and KS, whereas the incidence matrix only satisfies KS conditions. Similarly, every RS adjacency QPNFM is a KS adjacency QPNFM, but a KS adjacency QPNFM does not necessarily imply RS QPNFM. In this paper, we present an application of soft graphs in decision-making through the use of the adjacency matrix of a soft graph. We have developed an algorithm for this purpose and provide an example to demonstrate its application. [ABSTRACT FROM AUTHOR] more...

In this paper, we present equivalent characterizations of k-kernel symmetric (k-KS) Quadri Partitioned Neutrosophic Fuzzy Matrices (QPNFMs). Additionally, we establish the necessary and sufficient conditions for the Schur complement (SC) within a k-KS QPNFM to be k-symmetric. The study also offers equivalent characterizations of both KS and k-KS QPNFMs. A few fundamental examples of KS QPNFMs are provided to clarify these concepts. It is shown that although k-symmetry implies k-KS, the converse does not necessarily hold. Several fundamental properties of k-KS QPNFMs are also derived. Finally, decision-making model utilizing QPNSMs has been successfully developed and validated through its application to real-world problems. [ABSTRACT FROM AUTHOR] more...

This study explores a new type of matrix called a range-symmetric Fermatean neutrosophic fuzzy matrix (FNFM), inspired by the concept of range-Hermitian matrices. We demonstrate that all FNFMs inherently possess a specific property we term "Pythagorean neutrosophic fuzzy," (PNFM) but the reverse is not always true. Furthermore, we delve into graphical representations of FNFMs with specific symmetry properties (kernel-symmetric (KS), column symmetric, and range-symmetric (RS)) and show that these properties hold for all isomorphic graphs. The study goes on to establish equivalent characterizations for range-symmetric FNFMs and identify conditions for KS FNFMs. We introduce a novel concept: k-KS and RS FNFMs. Examples illustrate that KS FNFMs inherently possess k-KS, but not necessarily the other way around. This research contributes to a deeper understanding of symmetric FNFM and their potential applications, highlighting their importance in mathematical and computational fields. [ABSTRACT FROM AUTHOR] more...

This research examines an interval-valued secondary k-Range symmetric fuzzy matrix. It discusses the relationships between different types of matrices, specifically interval-valued s-k Range symmetric, interval-valued k-Range symmetric, and interval-valued Range symmetric matrices. The study establishes the necessary and sufficient criteria for an interval-valued s-k Range symmetric fuzzy matrix. It is demonstrated that s-symmetry implies s-Range symmetric and the reverse is necessarily true. Also, we illustrate a graphical representation of Kernel symmetric, Column symmetric, and Range symmetric adjacency and incidence fuzzy matrices. Every adjacency fuzzy matrix is symmetric, Range symmetric, Column symmetric, and Kernel symmetric, but the incidence matrix satisfies only the KS conditions. Similarly, every Range symmetric adjacency fuzzy matrix is a Kernel symmetric adjacency fuzzy matrix, but a Kernel symmetric adjacency fuzzy matrix need not be a Range symmetric fuzzy matrix. Also, in every isomorphic graph, its adjacency fuzzy matrix is a Kernel symmetric, Range symmetric, and Column symmetric adjacency fuzzy matrix, but the converse need not be true. Additionally, equivalent criteria for various g-inverses of an interval-valued s-k Range symmetric fuzzy matrix being interval-valued s-k Range symmetric matrices are also established. The generalized inverses of an interval-valued s-k Range symmetric matrix corresponding to the sets A{1,2}, A{1, 2, 3} and A{1, 2, 4} are characterized. In this paper, we present an application of soft graphs in decision-making through the use of the adjacency matrix of a soft graph. We have developed an algorithm for this purpose and provide an example to demonstrate its application. [ABSTRACT FROM AUTHOR] more...

In this paper, we introduce the concept of reverse sharp ordering on Intuitionistic Fuzzy matrix (IFM) as a special case of minus ordering. We also introduce the concept of reverse left-T and right-T orderings for IFM as an analogue of left-star and right-star partial orderings for complex matrices. Several properties of these ordering are derived. We show that these ordering preserve its Moore-penrose inverse property. Finally, we show that these ordering are identical for certain class of IFM. [ABSTRACT FROM AUTHOR] more...

The idea of Kernel and k-Kernel Symmetric (k-KS) Intuitionistic Fuzzy Matrices (IFM) are introduced with an example. We present some basic results of kernel symmetric matrices. We show that k-symmetric implies k-Kernel symmetric but the converse need not be true. The equivalent relations between kernel symmetric, k-kernel symmetric and Moore-Penrose inverse of IFM are explained with numerical results. [ABSTRACT FROM AUTHOR] more...

This paper introduces and explores the concept of secondary k-Range Symmetric (RS) Neutrosophic Fuzzy Matrices (NFM) and establishes its properties and relationships with other symmetric and secondary symmetric NFMs. The study defines secondary k-RS NFMs and provides insightful numerical examples to illustrate their characteristics. The paper investigates the interconnections among s-k-RS, s-RS, k-RS, and RS NFMs, discuss on their mutual relations. Additionally, the necessary and sufficient conditions for a given NFM to qualify as a s-k-RS NFM are identified. The research demonstrates that k-symmetry implies k-RS, and vice versa, contributing to a comprehensive understanding between different types of symmetries in NFMs. Graphical representations of RS, column symmetric, and kernel symmetric adjacency and incidence NFMs are presented, unveiling intriguing patterns and relationships. While every adjacency NFM is symmetric, range symmetric, column symmetric, and kernel symmetric, the incidence matrix satisfies only kernel symmetric conditions. The study further establishes that every range symmetric adjacency NFM is a kernel symmetric adjacency NFM, though the converse does not hold in general. The existence of multiple generalized inverses of NFMs in Fn is explored, with additional equivalent conditions for certain g-inverses of s-κ-RS NFMs to retain the s-κ-RS property. We conclude by characterizing the generalized inverses belonging to specific sets λ {1, 2}, λ {1, 2, 3}, and λ {1, 2, 4} of s-k-RS NFMs, providing a comprehensive framework for understanding the structure and properties of secondary k-Range Symmetric Neutrosophic Fuzzy Matrices. This research contributes to the mathematical literature by introducing a novel class of NFMs and establishing their fundamental properties and relationships, presenting new perspectives on matrix theory in the context of neutrosophic fuzzy logic. [ABSTRACT FROM AUTHOR] more...

Objective: The objective of this study is to establish the results of secondary k-column symmetric (CS) Neutrosophic fuzzy matrices. Methods and Findings: We have applied CS condition in neutrosophic environment to find the relation between s-k CS, s- CS, k- CS and CS. Novelty: We establish the necessary and sufficient criteria for s-k CS Neutrosophic fuzzy matrices and various g-inverses of an s - k CS Neutrosophic fuzzy matrices to be an s - k CS. The generalized inverses of an s - k CS P corresponding to the sets P{1, 2}, P{1, 2, 3} and P{1, 2, 4} are characterized. [ABSTRACT FROM AUTHOR] more...

In this article, First, we study the different orderings for k-idempotent Neutrosophic fuzzy matrices (NFM). With this idea, we also discover some properties for the k-Neutrosophic fuzzy matrices and demonstrate the connection between the generalized inverse and different orderings. We also go through some properties for the T-ordering, T-reverse ordering, minus, and space ordering in k-idempotent Neutrosophic fuzzy matrices using the g-inverses with numerical examples is given. Minus ordering is a partial ordering in the set of all regular fuzzy matrices. We have introduced ordering on k-idempotent fuzzy matrices and developed the theory of fuzzy matrix partial ordering. The minus ordering and k-space ordering are identical for k-idempotent matrices. Next, we introduce and study the concept of k-Idempotent Neutrosophic fuzzy matrix as a generalization of idempotent NFM via permutations. It is shown that a kidempotent NFM reduces to an idempotent NFM if and only if PK = KP. The Conditions for power symmetric NFM to be k-idempotent are derived and some related results are given. [ABSTRACT FROM AUTHOR] more...

The characterization of interval valued (IV) secondary k- range symmetric (RS) Neutrosophic fuzzy matrices have been examined in this study with an example. It is discussed how IV s-k RS, s- RS, IV k- RS, and IV RS matrices relate to one another. We establish the necessary and sufficient criteria for IV s-k RS Neutrosophic fuzzy matrices. The existence of several generalized inverses of a matrix in IV Neutrosophic fuzzy matrices. It is also established what are the equivalent criteria for various g-inverses of an IV s - κ RS fuzzy matrix to be an IV s - κ RS. The generalized inverses of an IV s - κ RS P corresponding to the sets P{1, 2}, P{1, 2, 3} and P{1, 2, 4} are characterized. [ABSTRACT FROM AUTHOR] more...

We develop the concept of range symmetric Neutrosophic Fuzzy Matrix and Kernel symmetric Neutrosophic Fuzzy Matrix analogous to that of an EP -matrix in the complex field. First we present equivalent characterizations of a range symmetric matrix and then derive equivalent conditions for a Neutrosophic Fuzzy Matrix to be kernel symmetric matrix and study the relation between range symmetric and kernel symmetric Neutrosophic Fuzzy Matrices. The idea of Kernel and k-Kernel Symmetric (k-KS) Neutrosophic Fuzzy Matrices (NFM) are introduced with an example. We present some basic results of kernel symmetric matrices. We show that k-symmetric implies k-Kernel symmetric but the converse need not be true. The equivalent relations between kernel symmetric, k-kernel symmetric and Moore-Penrose inverse of NFM are explained with numerical results. [ABSTRACT FROM AUTHOR] more...

In this paper, we introduce the concept of reverse sharp ordering on Neutrosophic Fuzzy matrix (NFM) as a special case of minus ordering. We also introduce the concept of reverse left-T and right-T orderings for NFM as an analogue of left-star and right-star partial orderings for complex matrices. Several properties of these ordering are derived. We show that these ordering preserve its Moore-penrose inverse property. Finally, we show that these ordering are identical for certain class of NFM. [ABSTRACT FROM AUTHOR] more...

In this article, we discuss various Inverses Minimum norm g-inverse, Least square g-inverse, Moore Penrose inverse, Group Inverse, Generalized Symmetric Neutrosophic Fuzzy Matrices. Also we describes secondary k-column symmetric Neutrosophic fuzzy matrices are produced. It is discussed how s-k-column symmetric, s-column symmetric, k-column symmetric, and column symmetric Neutrosophic fuzzy matrices relate to one another. For an Neutrosophic fuzzy matrices to be an s-k-column symmetric Neutrosophic fuzzy matrices, necessary and sufficient requirements are identified. [ABSTRACT FROM AUTHOR] more...

In this paper, first we shall define Pseudo Similarity for Neutrosophic Fuzzy Matrices and prove that Pseudo Similarity relation on pair of Neutrosophic Fuzzy Matrices. Also, we derive some relation between Pseudo Similarity and Idempotent matrices. Finally, we give in varies inverse of Neutrosophic Fuzzy Matrices. [ABSTRACT FROM AUTHOR] more...