DETERMINANT AND ADJOINT OF FUZZY NEUTROSOPHIC SOFT MATRICES

DETERMINANT AND ADJOINT OF FUZZY NEUTROSOPHIC SOFT MATRICES R.Uma∗, P. Murugadas†and S.Sriram‡ *† Department of Mathematics, Annamalai University, Annamalainagar-608002, India. ‡ Department of Mathematics, Mathematics wing, D.D.E, Annamalai University, Annamalainagar-608002, India.

Abstract In this paper, we have introduced the determinant and adjoint of a square Fuzzy Neutrosophic Soft Matrices (FNSMs). Also we define the circular FNSM and study some relations on square FNSM such as reflexivity, transitivity and circularity.

Keywords: Fuzzy Neutrosophic Soft Set, Fuzzy Neutrosophic Soft Matrix, Determinant of a square FNSM, Adjoint of a square FNSM.

1

Introduction

The theory of fuzzy set was introduced by Zadeh [?] as an appropriate mathematical instrument for description of uncertainty observed in nature. Since the inception it has got intensive acceptability in various fields. The traditional fuzzy sets is characterised by the membership value or the grade of membership value. Some times it may be very difficult to assign the membership value for fuzzy sets. Consequently the concept of interval valued fuzzy sets was proposed [?] to capture the uncertainty of grade of membership value. In some real life problems in expert system, belief ∗

uma83bala@gmail.com bodi muruga@yahoo.com ‡ ssm 3096@yahoo.co.in †

1

system, information fusion and so on, we must consider the truth membership as well as the falsity-membership for proper description of an object in uncertain, ambiguous environment. Neither the fuzzy sets nor the interval valued fuzzy sets is appropriate for such a situation. Intuitionistic fuzzy sets introduced by Atanassov [?] is appropriate for such a situation. The intuitionistic fuzzy sets can only handle the incomplete information considering both the truth membership (or simply membership) and falsity-membership (or non membership) values. It does not handle the indeterminate and inconsistent information which exists in belief system.

Smarandache [?] introduced the concept of neutrosophic set which is a Mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. The concept of soft set theory was introduced by Molodtsov [?] in 1999, it is a new approach for modeling vagueness and uncertainty. Soft set theory has a rich potential for applications in several directions, few of which had been shown by Molodtsov in his pioneer work. It is well known that the matrix formulation of a Mathematical formula gives extra advantages to handle the problem. The classical matrix theory cannot solve the problems involving various types of uncertainities.

That type of problems are solved by using fuzzy matrix[?]. Fuzzy matrix has been proposed to represent fuzzy relation in a system based on fuzzy set theory, Ovehinnikov [?].

Fuzzy matrices play an important role in Science and Technology. Kim [?, ?, ?] has explored some important result on the determinant of a square matrix. In Yong Yang and Chenli Ji [?], introduced a matrix representation of soft set and applied it in decision making problems. Rajarajeswari and Dhanalakshmi [?] introduced fuzzy soft matrix and its application in Medical diagnosis. Sumathi and Arockiarani [?] introduced new operations on fuzzy neutrosophic soft matrices. Mamouni Dhar [?] et al., have also deďŹ ned Neutrosophic fuzzy matrices and studied about square neutrosophic fuzzy matrices. In this article our main intention is to deďŹ ne determinant and adjoint of FNSMs. Furthermore, eorts have been made to establish some properties with the help of the new introduced deďŹ nition of determinant of square FNSMs. In section 1 we have introduced determinant of two types FNSM and its properties. In section 2, the deďŹ nition of adjoint of FNSM is given and some related Theorems are asserted.

2

preliminaries

DeďŹ nition 2.1. [?] Let U be an initial universe set and đ??¸ be a set of parameters.

Let P(U) denotes the power set of U. Consider a nonempty set đ??´, đ??´ ⊂ đ??¸. A pair (F,A) is called a soft set over U, where F is a mapping given by đ??š : đ??´ → đ?‘ƒ (đ?‘ˆ ).

2

DeďŹ nition 2.2. [?] Let đ?‘ˆ be an initial universe set and đ??¸ be a set of parameters.

Consider a non empty set đ??´, đ??´ ⊂ đ??¸. Let đ?‘ƒ (đ?‘ˆ ) denotes the set of all fuzzy neutrosophic sets of đ?‘ˆ . The collection (đ??š, đ??´) is termed to be the Fuzzy Neutrosophic Soft Set (FNSS) over đ?‘ˆ , Where F is a mapping given by đ??š : đ??´ → đ?‘ƒ (đ?‘ˆ ). Hereafter we simply consider đ??´ as FNSM over đ?‘ˆ instead of (đ??š, đ??´).

DeďŹ nition 2.3. [?] A neutrosophic set đ??´ on the universe of discourse đ?‘‹ is deďŹ ned as đ??´ = {â&#x;¨đ?‘Ľ, đ?‘‡đ??´ (đ?‘Ľ), đ??źđ??´ (đ?‘Ľ), đ??šđ??´ (đ?‘Ľ)â&#x;Š, đ?‘Ľ ∈ đ?‘‹}, where đ?‘‡, đ??ź, đ??š : đ?‘‹ →]− 0, 1+ [ and − 0 ≤ đ?‘‡đ??´ (đ?‘Ľ) + đ??źđ??´ (đ?‘Ľ) + đ??šđ??´ (đ?‘Ľ) ≤ 3+ .......(1) From philosophical point of view the neutrosophic set takes the value from real standard or non-standard subsets of ]− 0, 1+ [ . But in real life application especially in scientiďŹ c and Engineering problems it is diďŹƒcult to use neutrosophic set with value from real standard or non-standard subset of ]− 0, 1+ [ . Hence we consider the neutrosophic set which takes the value from the subset of [0, 1].

Therefore we can rewrite the equation (1) as 0 ≤ đ?‘‡đ??´ (đ?‘Ľ) + đ??źđ??´ (đ?‘Ľ) + đ??šđ??´ (đ?‘Ľ) ≤ 3.

In short an element đ?‘Ž Ëœ in the neutrosophic set đ??´ , can be written as đ?‘Ž Ëœ = â&#x;¨đ?‘Žđ?‘‡ , đ?‘Žđ??ź , đ?‘Žđ??š â&#x;Š, where đ?‘Žđ?‘‡ denotes degree of truth, đ?‘Žđ??ź denotes degree of indeterminacy, đ?‘Žđ??š denotes degree of falsity such that 0 ≤ đ?‘Žđ?‘‡ + đ?‘Žđ??ź + đ?‘Žđ??š ≤ 3.

Example 2.4. Assume that the universe of discourse đ?‘‹ = {đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 }, where đ?‘Ľ1 , đ?‘Ľ2 , and đ?‘Ľ3 characterises the quality, relaibility, and the price of the objects. It may be further assumed that the values of {đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 } are in [0, 1] and they are obtained from some investigations of some experts. The experts may impose their opinion in three components viz; the degree of goodness, the degree of indeterminacy and the degree of poorness to explain the characteristics of the objects. Suppose đ??´ is a Neutrosophic Set (NS) of đ?‘‹, such that đ??´ = {â&#x;¨đ?‘Ľ1 , 0.4, 0.5, 0.3â&#x;Š, â&#x;¨đ?‘Ľ2 , 0.7, 0.2, 0.4â&#x;Š, â&#x;¨đ?‘Ľ3 , 0.8, 0.3, 0.4â&#x;Š} , where for đ?‘Ľ1 the degree of goodness of quality is 0.4 , degree of indeterminacy of quality is 0.5 and degree of falsity of quality is 0.3 etc,.

Let ℹ�×� denotes FNSM of order � × � and ℹ� denotes FNSM of order � × �.

Operations on FNSM of type-I are deďŹ ned as follows. ⌊ âŒŞ ⌊ âŒŞ DeďŹ nition 2.5. [?] Let đ??´ = ( đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— ), đ??ľ = ( đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— ) ∈ â„ąđ?‘šĂ—đ?‘› . The componentwise addition multiplication is }deďŹ ned as { and componentwise } { } { đ??´ ⊕ đ??ľ = (đ?‘ đ?‘˘đ?‘? đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘ đ?‘˘đ?‘? đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— ). đ??´ ⊙ đ??ľ = (đ?‘–đ?‘›đ?‘“ {đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— }, đ?‘–đ?‘›đ?‘“ {đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— }).

DeďŹ nition 2.6. [?] Let đ??´ ∈ â„ąđ?‘šĂ—đ?‘› , đ??ľ ∈ â„ąđ?‘›Ă—đ?‘? , the composition of đ??´ and đ??ľ is deďŹ ned as 3

đ??´âˆ˜đ??ľ =

( đ?‘› ∑ đ?‘˜=1

( =

(đ?‘Žđ?‘‡đ?‘–đ?‘˜

∧

đ?‘?đ?‘‡đ?‘˜đ?‘— ),

đ?‘› ∑ (đ?‘Žđ??źđ?‘–đ?‘˜ ∧ đ?‘?đ??źđ?‘˜đ?‘— ), đ?‘˜=1

đ?‘› âˆ?

(đ?‘Žđ??šđ?‘–đ?‘˜ đ?‘˜=1

) ∨

đ?‘?đ??šđ?‘˜đ?‘— )

equivalently we can write the same as ) đ?‘› đ?‘› đ?‘› â‹ â‹ â‹€ (đ?‘Žđ?‘‡đ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— ), (đ?‘Žđ??źđ?‘–đ?‘˜ ∧ đ?‘?đ??źđ?‘˜đ?‘— ), (đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— ) .

đ?‘˜=1

đ?‘˜=1

đ?‘˜=1

The product đ??´ ∘ đ??ľ is deďŹ ned if and only if the number of columns of đ??´ is same as the number of rows of đ??ľ. đ??´ and đ??ľ are said to be conformable for multiplication.

We shall use đ??´đ??ľ instead of đ??´ ∘ đ??ľ.

DeďŹ nition 2.7. [?] Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) and đ?‘? ∈ â„ą = [0, 1]. DeďŹ ne the fuzzy ⌊ âŒŞ neutrosophic scalar multiplication as đ?‘?đ??´ = ( đ?‘–đ?‘›đ?‘“ {đ?‘?, đ?‘Žđ?‘‡đ?‘–đ?‘— }, đ?‘–đ?‘›đ?‘“ {đ?‘?, đ?‘Žđ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘?, đ?‘Žđ??šđ?‘–đ?‘— } ) ∈ â„ąđ?‘šĂ—đ?‘› .

For the universal matrix đ??˝1 , by the deďŹ nition 2.5 , đ?‘?đ??˝1 = đ?‘–đ?‘›đ?‘“ (đ?‘? ⊙ â&#x;¨1, 1, 0â&#x;Š) = (â&#x;¨đ?‘–đ?‘›đ?‘“ {đ?‘?, 1}, đ?‘–đ?‘›đ?‘“ {đ?‘?, 1}, đ?‘ đ?‘˘đ?‘?{đ?‘?, 0}â&#x;Š) = (â&#x;¨đ?‘?, đ?‘?, đ?‘?â&#x;Š) is the constant matrix all of whose entries are đ?‘?. Further under componentwise multiplication đ?‘?đ??˝1 ⊙ đ??´ = (â&#x;¨đ?‘?, đ?‘?, đ?‘?â&#x;Š) ⊙ (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) = (â&#x;¨đ?‘šđ?‘–đ?‘›{đ?‘?, đ?‘Žđ?‘‡đ?‘–đ?‘— }, đ?‘šđ?‘–đ?‘›{đ?‘?, đ?‘Žđ??źđ?‘–đ?‘— }, đ?‘šđ?‘Žđ?‘Ľ{đ?‘?, đ?‘Žđ??šđ?‘–đ?‘— }â&#x;Š) = đ?‘?đ??´ ......(2) DeďŹ nition 2.8. If đ??´ = (đ?‘Žđ?‘–đ?‘— ) ∈ â„ąđ?‘šĂ—đ?‘› , where (đ?‘Žđ?‘–đ?‘— ) = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š), then đ??´đ?‘? = (đ?‘?đ?‘–đ?‘— )đ?‘šĂ—đ?‘› where (đ?‘?đ?‘–đ?‘— ) = (â&#x;¨đ?‘Žđ??šđ?‘–đ?‘— , 1 − đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ?‘‡đ?‘–đ?‘— â&#x;Š), is the complement of đ??´.

DeďŹ nition 2.9. [?] The đ?‘› Ă— đ?‘š Zero matrix đ?‘‚1 is the matrix all of whose entries are of the form â&#x;¨0, 0, 1â&#x;Š. { â&#x;¨1, 1, 0â&#x;Š if đ?‘– = đ?‘— The đ?‘› Ă— đ?‘› identity matrix â„?1 is the matrix â„?1 = â&#x;¨0, 0, 1â&#x;Š if đ?‘– ∕= đ?‘— The đ?‘› Ă— đ?‘š universal matrix đ?’Ľ1 is the matrix all of whose entries are of the form â&#x;¨1, 1, 0â&#x;Š.

Operations on FNSM of type-II are deďŹ ned as follows.

DeďŹ nition 2.10. [?] Let đ??´ = (đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— ), đ??ľ = (đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— ) ∈ â„ąđ?‘šĂ—đ?‘› , the componentwise addition ⌊ and { componentwise } { multiplication } {is đ??šdeďŹ ned }âŒŞ as đ?‘‡ đ?‘‡ đ??ź đ??ź đ??š đ??´ ⊕ đ??ľ = ( đ?‘ đ?‘˘đ?‘? đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— ). ⌊ âŒŞ đ??´ ⊙ đ??ľ = ( đ?‘–đ?‘›đ?‘“ {đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— } ).

Analogous to FNSM of type-I we can deďŹ ne FNSM of type -II in the following way 4

DeďŹ nition 2.11. [?] Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) = (đ?‘Žđ?‘–đ?‘— ) ∈ â„ąđ?‘šĂ—đ?‘› and đ??ľ = (â&#x;¨đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— â&#x;Š) = (đ?‘?đ?‘–đ?‘— ) ∈ â„ąđ?‘›Ă—đ?‘? the product of đ??´ and đ??ľ is deďŹ ned as ) ( đ?‘› đ?‘› đ?‘› âˆ? âˆ? ∑⌊ âŒŞ âŒŞ ⌊ âŒŞ ⌊ đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— đ?‘Žđ??źđ?‘–đ?‘˜ ∨ đ?‘?đ??źđ?‘˜đ?‘— , đ??´âˆ—đ??ľ = đ?‘Žđ?‘‡đ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— ,

( =

đ?‘˜=1

đ?‘˜=1

đ?‘˜=1

equivalently we can write the same as ) đ?‘› đ?‘› đ?‘› â‹€ â‹€ â‹ âŒŞ âŒŞ ⌊ âŒŞ ⌊ ⌊ đ?‘‡ . đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— đ?‘Žđ??źđ?‘–đ?‘˜ ∨ đ?‘?đ??źđ?‘˜đ?‘— , đ?‘Žđ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— , đ?‘˜=1

đ?‘˜=1

đ?‘˜=1

the product đ??´ ∗ đ??ľ is deďŹ ned if and only if the number of columns of A is same as the number of rows of B. A and B are said to be conformable for multiplication.

DeďŹ nition 2.12. [?] The đ?‘›Ă—đ?‘š Zero matrix đ?‘‚2 is the matrix all { of whose entries are â&#x;¨1, 0, 0â&#x;Š if đ?‘– = đ?‘— of the form â&#x;¨0, 1, 1â&#x;Š. The đ?‘›Ă—đ?‘› identity matrix â„?2 is the matrix = â&#x;¨0, 1, 1â&#x;Š if đ?‘– ∕= đ?‘— The đ?‘› Ă— đ?‘š universal matrix đ?’Ľ2 is the matrix all of whose entries are of the form â&#x;¨1, 0, 0â&#x;Š.

DeďŹ nition 2.13. [?] Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) and đ?‘? ∈ â„ą, then the fuzzy neutrosophic scalar multiplication is deďŹ ned by đ?‘?đ??´ = (đ?‘–đ?‘›đ?‘“ {đ?‘?, đ?‘Žđ?‘‡đ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘?, đ?‘Žđ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘?, đ?‘Žđ??šđ?‘–đ?‘— }) Proposition 2.14. [?] If đ??´ ≤ đ??ľ , then đ??´đ??ś ≤ đ??ľđ??ś.

3

The determinant and adjoint of FNSM of type-I

DeďŹ nition 3.1. The determinant âˆŁđ??´âˆŁ of đ?‘› Ă— đ?‘› FNSM đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— .đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) is deďŹ ned as follows â‹ đ?‘‡ â‹€ đ??š â‹ đ??ź đ?‘Ž1đ?œŽ(1) ∨ ... ∨ đ?‘Žđ??šđ?‘›đ?œŽ(đ?‘›) â&#x;Š âˆŁđ??´âˆŁ = â&#x;¨ đ?‘Ž1đ?œŽ(1) ∧ ... ∧ đ?‘Žđ??źđ?‘›đ?œŽ(đ?‘›) , đ?‘Ž1đ?œŽ(1) ∧ ... ∧ đ?‘Žđ?‘‡đ?‘›đ?œŽ(đ?‘›) , đ?œŽâˆˆđ?‘†đ?‘›

đ?œŽâˆˆđ?‘†đ?‘›

đ?œŽâˆˆđ?‘†đ?‘›

where �� denotes the symmetric group of all permutations of the indices (1, 2, ...�).

Example 3.2. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) be a FNSM such that [ ] (0.5, 0.3, 0.4) (0.6, 0.7, 0.8) đ??´= (0.9, 0.6, 0.7) (0.5, 0.6, 0.7)

5

∣𝐴∣ = (⟨0.5, 0.3, 0.4⟩ ∧ ⟨0.5, 0.6, 0.7⟩) ∨ (⟨0.6, 0.7, 0.8⟩ ∧ ⟨0.9, 0.6, 0.7⟩) = ⟨0.5, 0.3, 0.7⟩ ∨ ⟨0.6, 0.6, 0.8⟩ = ⟨0.6, 0.6, 0.7⟩ Theorem 3.3. If a FNSM 𝐵 is obtained from an 𝑛 × 𝑛 FNSM 𝐴 = (⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩) by multiplying the 𝑖 − 𝑡ℎ row of A (𝑖 − 𝑡ℎ column) by 𝑘 ∈ [0, 1], then 𝐵 = 𝑘 ∣𝐴∣ .

Proof. Suppose that 𝐵 = (⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩), then ⋁

∣𝐵∣ = ⟨

𝜎∈𝑆𝑛

=⟨ ... ∨ 𝑘

⋁

𝑏𝑇1𝜎(1) ∧ ... ∧ 𝑏𝑇𝑛𝜎(𝑛) ,

⋁ 𝜎∈𝑆𝑛

𝑎𝑇1𝜎(1) ∧...∧𝑘 𝑎𝑖𝜎(𝑖) ∧...∧𝑎𝑇𝑛𝜎(𝑛) ,

𝜎∈𝑆𝑛 𝑎𝐹𝑖𝜎(𝑖) ...𝑎𝐹𝑛𝜎(𝑛) ⟩

= ⟨𝑘

𝑏𝐼1𝜎(1) ∧ ... ∧ 𝑏𝐼𝑛𝜎(𝑛) ,

⋁

𝑎𝑇1𝜎(1) ∧ ... ∧ 𝑎𝑇𝑛𝜎(𝑛) , 𝑘

𝜎∈𝑆𝑛 𝑇 𝑘⟨𝑎𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩

⋁ 𝜎∈𝑠𝑛

⋁ 𝜎∈𝑆𝑛

⋀ 𝜎∈𝑆𝑛

𝑏𝐹1𝜎(1) ∨ ... ∨ 𝑏𝐹𝑛𝜎(𝑛) ⟩

𝑎𝐼1𝜎(1) ∧...∧∧𝑘 𝑎𝐼𝑖𝜎(𝑖) ...∧...𝑎𝐼𝑛𝜎(𝑛) ,

⋀ 𝜎∈𝑆𝑛

𝑎𝐹1𝜎(1) ∨

𝑎𝐼1𝜎(1) ∧ ... ∧ 𝑎𝐼𝑛𝜎(𝑛) , 𝑘 ∧𝜎∈𝑆𝑛 𝑎𝐹1𝜎(1) ∨ ... ∨ 𝑎𝐹𝑛𝜎(𝑛) )

= = 𝑘 ∣𝐴∣.

Theorem 3.4. Let 𝐴 = (⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩) be an 𝑛 × 𝑛 FNSM then 𝑑𝑒𝑡(𝑃 𝐴) = 𝑑𝑒𝑡(𝐴) = 𝑑𝑒𝑡(𝐴𝑃 ), where 𝑃 is a permutation FNSM which is obtained from the identity FNSM by interchanging row i and row j.

Proof. Let 𝐴 = (⟨𝑐𝑇𝑖𝑗 , 𝑐𝐼𝑖𝑗 , 𝑐𝐹𝑖𝑗 ⟩). Then for any 𝑖, 𝑗, the 𝑖 − 𝑡ℎ(𝑗 − 𝑡ℎ) row of 𝑃 𝐴 is the 𝑗 − 𝑡ℎ(𝑖 − 𝑡ℎ respectively) row of 𝐴.

Infact, [ ] 𝑃 is a permutation FNSM which is generated by a permutation 𝑖 𝑗 𝑗 𝑖 .

[Since,] for any permutation 𝜎 ∈ 𝑆𝑛 , 𝑖 𝑗 𝜁 ∈ 𝑆𝑛 , 𝑗 𝑖 𝜎= ⋁ 𝑇 ⋁ 𝐼 ⋀ 𝐹 ∣𝑃 𝐴∣=⟨ 𝑐1𝜎(1) ∧ ... ∧ 𝑐𝑇𝑛𝜎(𝑛) , 𝑐1𝜎(1) ∧ ... ∧ 𝑐𝐼𝑛𝜎(𝑛) , 𝑐1𝜎(1) ∨ ... ∨ 𝑐𝐹𝑛𝜎(𝑛) ⟩ 𝜎∈𝑆𝑛 𝜎∈𝑆𝑛 𝜎∈𝑆𝑛 ⋁ 𝑇 ⋁ 𝐼 ⋀ 𝐹 =⟨ 𝑎1𝜁(1) ∧ ... ∧ 𝑎𝑇𝑛𝜁(𝑛) , 𝑎1𝜁(1) ∧ ... ∧ 𝑎𝐼𝑛𝜁(𝑛) , 𝑎1𝜁(1) ∨ ... ∨ 𝑎𝐹𝑛𝜁(𝑛) ⟩ 𝜁∈𝑆𝑛

𝜁∈𝑆𝑛

𝜁∈𝑆𝑛

= ∣𝐴∣ .

The case of 𝐴𝑃 is similar to the above proof.

6

DeďŹ nition 3.5. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š), be a đ?‘š Ă— đ?‘› FNSM then the transpose of đ??´ is deďŹ ned by, đ??´đ?‘‡ = (â&#x;¨đ?‘Žđ?‘‡đ?‘—đ?‘– , đ?‘Žđ??źđ?‘—đ?‘– , đ?‘Žđ??šđ?‘—đ?‘– â&#x;Š).

Theorem 3.6. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) be a FNSM then đ?‘‘đ?‘’đ?‘Ą(đ??´) = đ?‘‘đ?‘’đ?‘Ą(đ??´đ?‘‡ ), where đ??´đ?‘‡ denotes the transpose of đ??´.

Proof. Let đ??´đ?‘‡ = (â&#x;¨đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— â&#x;Š). Since each permutation đ?œŽ is one -to-one function, we

�have

â‹ đ??ź â‹€ đ??š đ?‘‡

đ??´ = â&#x;¨ â‹ đ?‘?đ?‘‡ đ?‘?1đ?œŽ(1) ∧ ... ∧ đ?‘?đ??źđ?‘›đ?œŽ(đ?‘›) , đ?‘?1đ?œŽ(1) ∨ ... ∨ đ?‘?đ??šđ?‘›đ?œŽ(đ?‘›) â&#x;Š 1đ?œŽ(1) ∧ ... ∧ đ?‘?đ?‘›đ?œŽ(đ?‘›) , đ?‘› đ?‘› đ?‘› â‹ đ?œŽâˆˆđ?‘† â‹ đ?œŽâˆˆđ?‘† â‹€ đ?œŽâˆˆđ?‘† =â&#x;¨ đ?‘Žđ?‘‡đ?œŽ(1)1 ∧ ... ∧ đ?‘Žđ?‘‡đ?œŽ(đ?‘›)đ?‘› , đ?‘Žđ??źđ?œŽ(1)1 ∧ ... ∧ đ?‘Žđ??źđ?œŽ(đ?‘›)đ?‘› , đ?‘Žđ??šđ?œŽ(1)1 ∨ ... ∨ đ?‘Žđ??šđ?œŽ(đ?‘›)đ?‘› â&#x;Š đ?œŽâˆˆđ?‘†đ?‘› đ?œŽâˆˆđ?‘†đ?‘› đ?œŽâˆˆđ?‘†đ?‘› â‹€ â‹ đ??ź â‹ đ?‘‡ đ?‘Žđ??š1đ?œ (1) ∨ ... ∨ đ?‘Žđ??šđ?‘›đ?œ (đ?‘›) â&#x;Š, đ?‘Ž1đ?œ (1) ∧ ... ∧ đ?‘Žđ??źđ?‘›đ?œ (đ?‘›) , =â&#x;¨ đ?‘Ž1đ?œ (1) ∧ ... ∧ đ?‘Žđ?‘‡đ?‘›đ?œ (đ?‘›) , đ?œ ∈đ?‘†đ?‘›

đ?œ ∈đ?‘†đ?‘›

đ?œ ∈đ?‘†đ?‘›

where the permutations đ?œ is induced by the rearrangement of each đ?œŽ in đ?‘†đ?‘› = âˆŁđ??´âˆŁ .

Theorem 3.7. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— ,đ??šđ?‘–đ?‘— â&#x;Š) be an đ?‘› Ă— đ?‘› FNSM. If đ??´ contains a zero row (column) then âˆŁđ??´âˆŁ = â&#x;¨0, 0, 1â&#x;Š.

Proof. Each term in âˆŁđ??´âˆŁ contains a factor of each row(column) and hence a factor of zero row (column). Thus each term of âˆŁđ??´âˆŁ is equal to zero, and consequently âˆŁđ??´âˆŁ = â&#x;¨0, 0, 1â&#x;Š. Hence zero means element of the form â&#x;¨0, 0, 1â&#x;Š.

Theorem 3.8. Let đ??´ = â&#x;¨(đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) be an đ?‘› Ă— đ?‘› FNSM, If đ??´ is triangular, then the determinant of đ??´, âŒŞ ⌊ â‹€ đ?‘‡ â‹€ đ??ź â‹ đ??š đ?‘Žđ?‘–đ?‘– đ?‘Žđ?‘–đ?‘– , âˆŁđ??´âˆŁ = đ?‘Žđ?‘–đ?‘– , 1≤đ?‘–≤đ?‘›

1≤�≤�

1≤�≤�

⌊ âŒŞ Proof. Suppose that đ??´ = (đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— ) is lower triangular. We consider the term of âˆŁđ??´âˆŁ thatâ‹€ â‹€ â‹ đ?‘Ąđ?‘Žđ?‘‡ = đ?‘Žđ?‘–đ?œŽ(đ?‘–) , đ?‘Ąđ?‘Žđ??ź = đ?‘Žđ?‘–đ?œŽ(đ?‘–) , đ?‘Ąđ?‘Žđ??š = đ?‘Žđ?‘–đ?œŽ(đ?‘–) .

1≤�≤�

1≤�≤�

1≤�≤�

Let đ?&#x153;&#x17D;(1) â&#x2C6;&#x2022;= 1. Then 1 < đ?&#x153;&#x17D;(1) and so đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;1đ?&#x153;&#x17D;(1) = 0, đ?&#x2018;&#x17D;đ??ź1đ?&#x153;&#x17D;(1) = 0, đ?&#x2018;&#x17D;đ??š1đ?&#x153;&#x17D;(1) = 1.

This means that đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2021; = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??ź = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??š = 1.

If đ?&#x153;&#x17D;(1) â&#x2C6;&#x2022;= 1. Now, let đ?&#x153;&#x17D;(1) = 1 and đ?&#x153;&#x17D;(2) â&#x2C6;&#x2022;= 2 then 2 < đ?&#x153;&#x17D;(2) and đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;2đ?&#x153;&#x17D;(2) = 0, đ?&#x2018;&#x17D;đ??ź2đ?&#x153;&#x17D;(2) = 0, đ?&#x2018;&#x17D;đ??š2đ?&#x153;&#x17D;(2) = 1, and đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2021; = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??ź = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??š = 1, This means that đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2021; = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??ź = 0, đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??š = 1, if đ?&#x153;&#x17D;(1) â&#x2C6;&#x2022;= 1 or đ?&#x153;&#x17D;(2) â&#x2C6;&#x2022;= 2.

Therefore, in this method, we know that each of terms đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2021; , đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??ź , đ?&#x2018;Ąđ?&#x2018;&#x17D;đ??š , for đ?&#x153;&#x17D;(1) â&#x2C6;&#x2022;= 1, đ?&#x153;&#x17D;(2) â&#x2C6;&#x2022;= 2...đ?&#x153;&#x17D;(đ?&#x2018;&#x203A;) â&#x2C6;&#x2022;= đ?&#x2018;&#x203A; must be zero, zero, one respectively, Consequently, 7

( â&#x2C6;Łđ??´â&#x2C6;Ł =

â&#x;¨

â&#x2039;&#x20AC;

đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; ,

1â&#x2030;¤đ?&#x2018;&#x2013;â&#x2030;¤đ?&#x2018;&#x203A;

đ?&#x2018;&#x17D;đ??źđ?&#x2018;&#x2013;đ?&#x2018;&#x2013; ,

â&#x2039;&#x20AC; 1â&#x2030;¤đ?&#x2018;&#x2013;â&#x2030;¤đ?&#x2018;&#x203A;

â&#x2039;

) đ?&#x2018;&#x17D;đ??šđ?&#x2018;&#x2013;đ?&#x2018;&#x2013; â&#x;Š .

1â&#x2030;¤đ?&#x2018;&#x2013;â&#x2030;¤đ?&#x2018;&#x203A;

The following theorem is evident from the deďŹ nition of determinant of FNSM. Theorem 3.9. Let đ??´ and đ??ľ be two FNSM. Then â&#x2C6;Łđ??´đ??ľâ&#x2C6;Ł â&#x2030;Ľ â&#x2C6;Łđ??´â&#x2C6;Ł â&#x2C6;Łđ??ľâ&#x2C6;Ł .

4

The Adjoint of FNSM

DeďŹ nition 4.1. The adjoint of an đ?&#x2018;&#x203A; Ă&#x2014; đ?&#x2018;&#x203A; FNSM đ??´ denoted by đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´, is deďŹ ned as follows đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = â&#x2C6;Łđ??´đ?&#x2018;&#x2014;đ?&#x2018;&#x2013; â&#x2C6;Ł is the determinant of the (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1)Ă&#x2014;(đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) FNSM formed by deleting row đ?&#x2018;&#x2014; and column đ?&#x2018;&#x2013; from đ??´ and đ??ľ = đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´.

Remark 4.2. We can write the element đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; of đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ = đ??ľ = (đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; ) as follows: â&#x152;Ş â&#x2C6;&#x2018; â&#x2C6;? â&#x152;Š đ?&#x2018;&#x2021; đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;&#x17D;đ??źđ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;&#x17D;đ??šđ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = đ?&#x153;&#x2039;â&#x2C6;&#x2C6;đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; đ?&#x2018;&#x203A;đ?&#x2018;&#x2013; đ?&#x2018;Ąâ&#x2C6;&#x2C6;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014;

where đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; = {1, 2, 3.....đ?&#x2018;&#x203A;} â&#x2C6;&#x2013; {đ?&#x2018;&#x2014;} and đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; đ?&#x2018;&#x203A;đ?&#x2018;&#x2013; is the set of all permutation of set đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; over the set đ?&#x2018;&#x203A;đ?&#x2018;&#x2013; . [ ] â&#x;¨0.2, 0.5, 0â&#x;Š â&#x;¨0.2, 0.3, 0.5â&#x;Š Example 4.3. Let đ??´ = , â&#x;¨0.6, 0.2, 0.3â&#x;Š â&#x;¨0.6, 0.7, 0.3â&#x;Š [ ] đ??´ đ??´ then đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ = đ??´11 đ??´21 12

22

= â&#x;¨0.6, 0.7, 0.3â&#x;Š = â&#x;¨0.6, 0.2, 0.3â&#x;Š = â&#x;¨0.2, 0.3, 0.5â&#x;Š = â&#x;¨0.2, [ 0.5, 0â&#x;Š ] â&#x;¨0.6, 0.7, 0.3â&#x;Š â&#x;¨0.2, 0.3, 0.5â&#x;Š đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ = â&#x;¨0.6, 0.2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0â&#x;Š

đ??´11 đ??´12 đ??´21 đ??´22

Proposition 4.4. For đ?&#x2018;&#x203A; Ă&#x2014; đ?&#x2018;&#x203A; FNSM đ??´ and đ??ľ we have the following 1. đ??´ â&#x2030;¤ đ??ľ implies đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ â&#x2030;¤ đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??ľ, 2. đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ + đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??ľ â&#x2030;¤ đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;(đ??´ + đ??ľ), 3. đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´đ?&#x2018;&#x2021; = (đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)đ?&#x2018;&#x2021; .

Proof. (i) = đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??ľ. That is â&#x2C6;&#x2018; Letâ&#x2C6;?đ??ś =đ?&#x2018;&#x2021; đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´đ??ź and đ??ˇ đ??š đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = â&#x;¨đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) â&#x;Š and đ?&#x153;&#x2039;â&#x2C6;&#x2C6;đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; đ?&#x2018;&#x203A;đ?&#x2018;&#x2013; đ?&#x2018;Ąâ&#x2C6;&#x2C6;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; â&#x2C6;&#x2018; â&#x2C6;? đ?&#x2018;&#x2021; đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = â&#x;¨đ?&#x2018;?đ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;?đ??źđ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) , đ?&#x2018;?đ??šđ?&#x2018;Ąđ?&#x153;&#x2039;(đ?&#x2018;Ą) â&#x;Š. đ?&#x153;&#x2039;â&#x2C6;&#x2C6;đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014; đ?&#x2018;&#x203A;đ?&#x2018;&#x2013; đ?&#x2018;Ąâ&#x2C6;&#x2C6;đ?&#x2018;&#x203A;đ?&#x2018;&#x2014;

8

It is clear that 𝑐𝑖𝑗 ≤ 𝑑𝑖𝑗 since 𝑎𝑇𝑡𝜋(𝑡) ≤ 𝑏𝑇𝑡𝜋(𝑡) 𝑎𝐼𝑡𝜋(𝑡) ≤ 𝑏𝐼𝑡𝜋(𝑡) 𝑎𝐹𝑡𝜋(𝑡) ≥ 𝑏𝐹𝑡𝜋(𝑡) for every 𝑡 ∕= 𝑗, 𝜋(𝑡) ∕= 𝑖. (ii) Since 𝐴 , 𝐵 ≤ 𝐴 + 𝐵, it is clear that 𝑎𝑑𝑗𝐴, 𝑎𝑑𝑗𝐵 ≤ 𝑎𝑑𝑗(𝐴 + 𝐵) and so 𝑎𝑑𝑗𝐴 + 𝑎𝑑𝑗𝐵 ≤ 𝑎𝑑𝑗(𝐴 + 𝐵). (iii)Let = 𝑎𝑑𝑗𝐴 and 𝐶 = 𝑎𝑑𝑗𝐴𝑇 . Then ∑ 𝐵∏ 𝑏𝑖𝑗 = ⟨𝑎𝑇𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩ and 𝜋∈𝑆𝑛𝑗 𝑛𝑖 𝑡∈𝑛𝑗 ∑ ∏ 𝑐𝑖𝑗 = ⟨𝑎𝑇𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩, 𝜋∈𝑆𝑛𝑗 𝑛𝑖 𝜋(𝑡)∈𝑛𝑗

which is the element 𝑏𝑗𝑖 . Hence (𝑎𝑑𝑗𝐴)𝑇 = 𝑎𝑑𝑗𝐴𝑇 .

Proposition 4.5. Let 𝐴 be an 𝑛 × 𝑛 FNSM. Then (1) 𝐴 𝑎𝑑𝑗𝐴 ≥ ∣𝐴∣ 𝐼𝑛 , (2) (𝑎𝑑𝑗𝐴)𝐴 ≥ ∣𝐴∣ 𝐼𝑛 .

Proof. (1) Let 𝐶 = 𝐴 𝑎𝑑𝑗𝐴. The i-th row of 𝐴 is (⟨𝑎𝑇𝑖1 , 𝑎𝐼𝑖1 , 𝑎𝐹𝑖1 ⟩ ⟨𝑎𝑇𝑖2 , 𝑎𝐼𝑖2 , 𝑎𝐹𝑖2 ⟩..., ⟨𝑎𝑇𝑖𝑛 , 𝑎𝐼𝑖𝑛 , 𝑎𝐹𝑖𝑛 ⟩). By the definition of 𝑎𝑑𝑗𝐴, the j-th column of 𝑎𝑑𝑗𝐴 is given by (∣𝐴𝑗1 ∣∑ , ∣𝐴𝑗2 ∣ , ..., ∣𝐴𝑗𝑛 ∣)𝑇 . So that ∑ 𝑇 𝐼 𝐹 ⟨𝑐𝑖𝑗 , 𝑐𝑖𝑗 , 𝑐𝑖𝑗 ⟩ = 𝑛𝑘=1 ⟨𝑎𝑇𝑖𝑘 , 𝑎𝐼𝑖𝑘 , 𝑎𝐹𝑖𝑘 ⟩ ∣𝐴𝑗𝑘 ∣ ≥ 0 and hence ⟨𝑐𝑇𝑖𝑖 , 𝑐𝐼𝑖𝑖 , 𝑐𝐹𝑖𝑖 ⟩ = 𝑛𝑘=1 ⟨𝑎𝑇𝑖𝑘 , 𝑎𝐼𝑖𝑘 , 𝑎𝐹𝑖𝑘 ⟩ ∣𝐴𝑖𝑘 ∣ which is equal to ∣𝐴∣. Thus 𝐶 = 𝐴𝑎𝑑𝑗𝐴 ≥ ∣𝐴∣ 𝐼𝑛 .

(2) The proof is similar to (1).

Proposition 4.6. If a FNSM matrix 𝐴 has a zero row then (𝑎𝑑𝑗𝐴)𝐴 = (⟨0, 0, 1⟩) (the zero matrix). ∑ Proof. Let 𝐻 = (𝑎𝑑𝑗𝐴)𝐴. That is, ℎ𝑖𝑗 = ∣𝐴𝑘𝑖 ∣ ⟨𝑎𝑇𝑘𝑗 𝑎𝐼𝑘𝑗 𝑎𝐹𝑘𝑗 ⟩. If the i-th row of 𝐴 𝑘

is zero, that means (⟨0, 0, 1⟩) , then 𝐴𝑘𝑖 contains a zero row where 𝑘 ∕= 𝑖 and so ∣𝐴𝑘𝑖 ∣ = ⟨0, 0, 1⟩ (by the Theorem 3.7) for if 𝑘 = 𝑖, then 𝑎𝑖𝑗 = 0 for every j and hence ∑ every 𝑘𝑇 ∕= 𝑖𝐼 and 𝐹 ∣𝐴𝑘𝑖 ∣ ⟨𝑎𝑘𝑗 , 𝑎𝑘𝑗 , 𝑎𝑘𝑗 ⟩ = (⟨0, 0, 1⟩). 𝑘

Thus (𝑎𝑑𝑗𝐴)𝐴 = (⟨0, 0, 1⟩)

9

Theorem 4.7. For a FNSM 𝐴 we have ∣𝐴∣ = ∣𝑎𝑑𝑗𝐴∣ . ∣𝐴11 ∣ ∣𝐴21 ∣ ⋅ ⋅ ⋅ ⎢ ∣𝐴12 ∣ ∣𝐴22 ∣ ⋅ ⋅ ⋅ Proof. Since 𝑎𝑑𝑗𝐴 = ⎣ .. .. .. . . . ∣𝐴1𝑛 ∣ ∣𝐴2𝑛 ∣ ⋅ ⋅ ⋅ we have ∑

𝐴1𝜋(1)

𝐴2𝜋(2) ... 𝐴𝑛𝜋(𝑛)

∣𝑎𝑑𝑗𝐴∣ = ⎡

⎤ ∣𝐴𝑛1 ∣ ∣𝐴𝑛2 ∣ ⎥ .. ⎦ . ∣𝐴𝑛𝑛 ∣

𝜋∈𝑆𝑛 𝑛

∑ ∏

𝐴𝑖𝜋(𝑖)

=

=

𝜋∈𝑆𝑛 𝑖=1 𝑛 ∑ ∏

[

∑

(

∏

𝜋∈𝑆𝑛 𝑖=1 𝜋∈𝑆𝑛𝑖 𝑛𝜋(𝑖) 𝑡∈𝑛𝑖

⟨𝑎𝑇𝑡𝜃(𝑡) , 𝑎𝐼𝑡𝜃(𝑡) , 𝑎𝐹𝑡𝜃(𝑡) ⟩)]

∑ ∏ 𝑇 ∑ ∏ 𝑇 [( ⟨𝑎𝑡𝜃(𝑡) , 𝑎𝐼𝑡𝜃(𝑡) , 𝑎𝐹𝑡𝜃(𝑡) ⟩)( ⟨𝑎𝑡𝜃(𝑡) , 𝑎𝐼𝑡𝜃(𝑡) , 𝑎𝐹𝑡𝜃(𝑡) ⟩)... 𝜋∈𝑆𝑛 𝜋∈𝑆𝑛1 𝑛𝜋(1) 𝑡∈𝑛1 𝜋∈𝑆𝑛2 𝑛𝜋(2) 𝑡∈𝑛2 ∏ 𝑇 ∑ ⟨𝑎𝑡𝜃(𝑡) , 𝑎𝐼𝑡𝜃(𝑡) , 𝑎𝐹𝑡𝜃(𝑡) ⟩)] ( ∑

=

𝜋∈𝑆𝑛𝑛 𝑛𝜋(𝑛) 𝑡∈𝑛𝑛

∑

=

[(

𝜋∈𝑠𝑛

∏

𝑡∈𝑛1

⟨𝑎𝑇𝑡𝜃1 (𝑡) , 𝑎𝐼𝑡𝜃1 (𝑡) , 𝑎𝐹𝑡𝜃𝑛 (𝑡) ⟩)(

∏

𝑡∈𝑛1

⟨𝑎𝑇𝑡𝜃2 (𝑡) , 𝑎𝐼𝑡𝜃2 (𝑡) , 𝑎𝐹𝑡𝜋2 (𝑡) ⟩)(

∏

𝑡∈𝑛1

⟨𝑎𝑇𝑡𝜃𝑛 (𝑡) , 𝑎𝐼𝑡𝜃𝑛 (𝑡) , 𝑎𝐹𝑡𝜃𝑛 (𝑡) ⟩)]

∑ for some 𝜃1 ∈ 𝑆𝑛1 𝑛𝜋(1) , 𝜃2 ∈ 𝑆𝑛2 𝑛𝜋(1) , ...𝜃𝑛 ∈ 𝑆𝑛𝑛 𝑛𝜋(𝑛) = [(𝑎2𝜃1 (2) 𝑎3𝜃1 (3) ...𝑎𝑛𝜃1 (𝑛) )(𝑎1𝜃2 (1) 𝑎3𝜃2 (3) ...𝑎𝑛𝜃2 (𝑛) )...(𝑎1𝜃𝑛 (1) 𝑎2𝜃𝑛 (2) ...𝑎𝑛−1𝜃𝑛 (𝑛−1) )] 𝜋∈𝑆 𝑛 ∑ = [(𝑎1𝜃2 (1) 𝑎1𝜃3 (1) ...𝑎1𝜃𝑛 (1) )(𝑎2𝜃1 (2) 𝑎2𝜃1 (2) ...𝑎2𝜃𝑛 (2) )(𝑎3𝜃1 (3) 𝑎3𝜃2 (3) 𝑎3𝜃4 (3) 𝜋∈𝑆𝑛

...𝑎3𝜃 ∑𝑛 (3) )...(𝑎𝑛𝜃1 (𝑛) 𝑎𝑛𝜃2 (𝑛) ...𝑎𝑛𝜃𝑛 (𝑛) )] = [(𝑎1𝜃𝑓1 (1) 𝑎2𝜃𝑓2 (2) ...𝑎𝑛𝜃𝑓𝑛 (𝑛) ] for some 𝑓ℎ ∈ {1, 2, ..., 𝑛} ∖ {ℎ}, ℎ = 1, 2, ..., 𝑛. How𝜋∈𝑆𝑛

ever because ∑ 𝑎ℎ𝜃𝑓ℎ (ℎ) ∕= 𝑎ℎ𝜎 (𝑓ℎ ). We can see that 𝑎ℎ 𝜃𝑓ℎ ℎ = 𝑎ℎ𝜋 (𝑓ℎ ) therefore, ∣𝑎𝑑𝑗𝐴∣ = 𝑎1𝜎(1) 𝑎2𝜎(2) ...𝑎𝑛𝜎(𝑛) , 𝜎∈𝑆𝑛

which is the expansion of ∣𝐴∣. This complete the proof.

Definition 4.8. An 𝑚×𝑛 FNSM 𝐴 is said to be constant if ⟨𝑎𝑇𝑖𝑘 , 𝑎𝐼𝑖𝑘 , 𝑎𝐹𝑖𝑘 ⟩ = ⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩ for all 𝑖, 𝑗, 𝑘, that is its row are equal to each other.

Proposition 4.9. Let 𝐴 be an 𝑛 × 𝑛 constant FNSM Then we have: (1). (𝑎𝑑𝑗𝐴)𝑇 is constant, (2). 𝐶 = 𝐴(𝑎𝑑𝑗𝐴) is constant and 𝐶𝑖𝑗 = ∣𝐴∣ which is the least element in 𝐴.

Proof. (1)Let ∑ ∏𝐵 = 𝑇𝑎𝑑𝑗𝐴.𝐼 Then𝐹 𝑏𝑖𝑗 = (⟨𝑎𝑡𝜋(𝑡) , 𝑎𝑡𝜋(𝑡) , 𝑎𝑡𝜋(𝑡) ⟩) and 𝑏𝑖𝑘 = 𝜋∈𝑆𝑛𝑗 𝑛𝑖 𝑡∈𝑛𝑗

10

∑

∏

𝜋∈𝑆𝑛𝑘 𝑛𝑖 𝑡∈𝑛𝑘

(⟨𝑎𝑇𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩).

We notice that đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; since the numbers đ?&#x153;&#x2039;(đ?&#x2018;Ą) of columns cannot be changed in the two expansion of đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; and đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; . So that (đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)đ?&#x2018;&#x2021; is constant.

(2) Since đ??´ is constant we can see that đ??´đ?&#x2018;&#x2014;đ?&#x2018;&#x2DC; = đ??´đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; and so â&#x2C6;Łđ??´đ?&#x2018;&#x2014;đ?&#x2018;&#x2DC; â&#x2C6;Ł = â&#x2C6;Łđ??´đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; â&#x2C6;Ł for every đ?&#x2018;&#x2013;, đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; {1, 2, ..., đ?&#x2018;&#x203A;}. Thus đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = =

đ?&#x2018;&#x203A; â&#x2C6;&#x2018; đ?&#x2018;&#x2DC;=1 đ?&#x2018;&#x203A; â&#x2C6;&#x2018;

(â&#x;¨đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; , đ?&#x2018;&#x17D;đ??źđ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; , đ?&#x2018;&#x17D;đ??šđ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; â&#x;Š) â&#x2C6;Łđ??´đ?&#x2018;&#x2014;đ?&#x2018;&#x2DC; â&#x2C6;Ł (â&#x;¨đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; , đ?&#x2018;&#x17D;đ??źđ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; , đ?&#x2018;&#x17D;đ??šđ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; â&#x;Š) â&#x2C6;Łđ??´đ?&#x2018;&#x2013;đ?&#x2018;&#x2DC; â&#x2C6;Ł

đ?&#x2018;&#x2DC;=1

= â&#x2C6;Łđ??´â&#x2C6;Ł .

Now, â&#x2C6;&#x2018; â&#x2C6;Łđ??´â&#x2C6;Ł = đ?&#x2018;&#x17D;1đ?&#x153;&#x2039;(1) đ?&#x2018;&#x17D;2đ?&#x153;&#x2039;(2) ...đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x153;&#x2039;(đ?&#x2018;&#x203A;) đ?&#x153;&#x2039;â&#x2C6;&#x2C6;đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;

= đ?&#x2018;&#x17D;2đ?&#x153;&#x2039;(2) đ?&#x2018;&#x17D;3đ?&#x153;&#x2039;(3) ...đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x153;&#x2039;(đ?&#x2018;&#x203A;) for any đ?&#x153;&#x2039; â&#x2C6;&#x2C6; đ?&#x2018;&#x2020;đ?&#x2018;&#x203A; (since đ??´ is constant). Taking đ?&#x153;&#x2039; as the identity permutation we get â&#x2C6;Łđ??´â&#x2C6;Ł = đ?&#x2018;&#x17D;11 đ?&#x2018;&#x17D;22 ...đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; which is the least element in đ??´.

DeďŹ nition 4.10. For a FNSM đ??´ â&#x2C6;&#x2C6; â&#x201E;ąđ?&#x2018;&#x203A;Ă&#x2014;đ?&#x2018;&#x203A; we have the following (1) If đ??´ â&#x2030;Ľ đ??źđ?&#x2018;&#x203A; , then đ??´ is called reďŹ&#x201A;exive.

(2) If đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; â&#x2030;Ľ đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014;, then đ??´ is called weakely reďŹ&#x201A;exive for all đ?&#x2018;&#x2013;, đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; {1, 2, ..., đ?&#x2018;&#x203A;} where đ??´ = (đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; ) = (â&#x;¨đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; , đ?&#x2018;&#x17D;đ??źđ?&#x2018;&#x2013;đ?&#x2018;&#x2014; , đ?&#x2018;&#x17D;đ??šđ?&#x2018;&#x2013;đ?&#x2018;&#x2014; â&#x;Š) (3)If đ??´ = đ??´đ?&#x2018;&#x2021; , then đ??´ is called symmetric (4)If đ??´ = đ??´2 , then đ??´ is called idempotent (5)If đ??´2 â&#x2030;¤ đ??´, then đ??´ is called transitive.

Proposition 4.11. Let đ??´ be an đ?&#x2018;&#x203A; Ă&#x2014; đ?&#x2018;&#x203A; reďŹ&#x201A;exive FNSM. Then đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ = đ??´đ?&#x2018;&#x2DC; where đ??´đ?&#x2018;&#x2DC; is idempotent and đ?&#x2018;&#x2DC; â&#x2030;¤ đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1.

Proof. The proof of the proposition is similar to fuzzy matrices refer[?].

Proposition 4.12. Let đ??´ be an đ?&#x2018;&#x203A; Ă&#x2014; đ?&#x2018;&#x203A; reďŹ&#x201A;exive FNSM. Then we have the following: (1)đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´2 = (đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)2 = đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´, (2)If đ??´ is idempotent, then đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ = đ??´, (3)đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ is reďŹ&#x201A;exive, (4)đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) = đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´, (5)đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´ â&#x2030;Ľ đ??´, (6)đ??´(đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)đ??´ = đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´.

11

Proof. (1) Since 𝐴 is reflexive, we get 𝐴2 is also reflexive and 𝑎𝑑𝑗𝐴2 = (𝐴2 )𝑘 = (𝐴𝑘 )2 = (𝑎𝑑𝑗𝐴)2 . But since 𝐴𝑘 is idempotent, we have (𝑎𝑑𝑗𝐴)2 = (𝑎𝑑𝑗𝐴).

(2)We have by proposition 2.10 that 𝑎𝑑𝑗𝐴 = 𝐴𝑘 (𝑘 ≤ 𝑛 − 1). But we have also that 𝐴 is idempotent. So 𝐴𝑘 = 𝐴. Thus 𝑎𝑑𝑗𝐴 = 𝐴.

(3) Let 𝐵 =∑𝑎𝑑𝑗𝐴. is, ∏ That 𝑇 = ⟨𝑎𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩.

(⟨𝑏𝑇𝑖𝑖 𝑏𝐼𝑖𝑖 𝑏𝐹𝑖𝑖 ⟩)

𝜋∈𝑆𝑛𝑖 𝑡∈𝑛𝑖

Taking the identity permutation 𝜋(𝑡) = 𝑡 we get (⟨𝑏𝑇𝑖𝑖 , 𝑏𝑇𝑖𝑖 , 𝑏𝑇𝑖𝑖 ⟩) ≥ ⟨𝑎𝑇11 𝑎𝑇22 ...𝑎𝑇𝑖−1𝑖−1 𝑎𝑇𝑖+1𝑖+1 ...𝑎𝑇𝑛𝑛 , 𝑎𝐼11 𝑎𝐼22 ...𝑎𝐼𝑖−1𝑖−1 𝑎𝐼𝑖+1𝑖+1 ...𝑎𝐼𝑛𝑛 , 𝑎𝐹11 𝑎𝐹22 ...𝑎𝐹𝑖−1𝑖−1 𝑎𝐹𝑖+1𝑖+1 ...𝑎𝐹𝑛𝑛 ⟩ = (⟨1, 1, 0⟩) that is ⟨𝑏𝑇𝑖𝑖 , 𝑏𝐼𝑖𝑖 , 𝑏𝐹𝑖𝑖 ⟩ = ⟨1, 1, 0⟩ and 𝑎𝑑𝑗𝐴 is thus reflexive.

(4) Since 𝐴 is reflexive, we get 𝑎𝑑𝑗𝐴 is idempotent by the above proposition and reflexive by (3). So that by (2) 𝑎𝑑𝑗(𝑎𝑑𝑗𝐴) = 𝑎𝑑𝑗𝐴.

(5) Let 𝐵 = 𝑎𝑑𝑗𝐴. ∑ That ∏ is𝑇 ⟨𝑎𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩. (⟨𝑏𝑇𝑖𝑗 𝑏𝐼𝑖𝑗 𝑏𝐹𝑖𝑗 ⟩) = 𝜋∈𝑆𝑛𝑗 𝑛𝑖 𝑡∈𝑛𝑗

Taking the identity permutation 𝜋(ℎ) = ] ℎ, 𝜋(𝑖) = 𝑗, ℎ ∕= 𝑖, that is the permutation [ 1 2 3 ⋅⋅⋅ 𝑖 ⋅⋅⋅𝑗 − 1 𝑗 + 1⋅⋅⋅𝑛 1 2 3 ⋅⋅⋅ 𝑗 ⋅⋅⋅𝑗 − 1 𝑗 + 1⋅⋅⋅𝑛 then ⟨𝑎𝑇11 𝑎𝑇22 ...𝑎𝑇𝑖−1𝑖−1 𝑎𝑇𝑖+1𝑖+1 ...𝑎𝑇𝑛𝑛 , 𝑎𝐼11 𝑎𝐼22 ...𝑎𝐼𝑖−1𝑖−1 𝑎𝐼𝑖+1𝑖+1 ...𝑎𝐼𝑛𝑛 , 𝑎𝐹11 𝑎𝐹22 ...𝑎𝐹𝑖−1𝑖−1 𝑎𝐹𝑖+1𝑖+1 ...𝑎𝐹𝑛𝑛 ⟩ is a term of ⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩. So that ⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩ ≥ ⟨𝑎𝑇11 , 𝑎𝑇22 ...𝑎𝑇𝑖−1𝑖−1 𝑎𝑇𝑖+1𝑖+1 ...𝑎𝑇𝑛𝑛 = ⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩.

Therefore 𝐵 = 𝑎𝑑𝑗𝐴 ≥ 𝐴.

(6) Let 𝐶 = 𝐴(𝑎𝑑𝑗𝐴) and 𝐷 = (𝑎𝑑𝑗𝐴)𝐴. Then 𝑛 ∑ ⟨𝑎𝑇𝑖𝑘 , 𝑎𝐼𝑖𝑘 , 𝑎𝐹𝑖𝑘 ⟩ ∣𝐴𝑗𝑘 ∣ ⟨𝑐𝑇𝑖𝑗 , 𝑐𝐼𝑖𝑗 , 𝑐𝐹𝑖𝑗 ⟩ = 𝑘=1

≥

⟨𝑎𝑇𝑖𝑖 , 𝑎𝐼𝑖𝑖 , 𝑎𝐹𝑖𝑖 ⟩ ∣𝐴𝑗𝑖 ∣

= ∣𝐴𝑗𝑖 ∣ = (⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩) and 𝑑𝑖𝑗 =

𝑛 ∑

∣𝐴𝑘𝑖 ∣ ⟨𝑎𝑇𝑘𝑗 , 𝑎𝐼𝑘𝑗 , 𝑎𝐹𝑘𝑗 ⟩

𝑘=1

≥ ∣𝐴𝑗𝑖 ∣ ⟨𝑎𝑇𝑗𝑗 , 𝑎𝐼𝑗𝑗 , 𝑎𝐹𝑗𝑗 ⟩ = ∣𝐴𝑗𝑖 ∣ = (⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩).

Thus we have 𝐴(𝑎𝑑𝑗𝐴) ≥ 𝑎𝑑𝑗𝐴 and (𝑎𝑑𝑗𝐴)𝐴 ≥ 𝑎𝑑𝑗𝐴. But by (1) and (5) and proposition [4.11] we see that 𝑎𝑑𝑗𝐴 = (𝑎𝑑𝑗𝐴)(𝑎𝑑𝑗𝐴) ≥ 𝐴𝑎𝑑𝑗𝐴. So that 𝐴(𝑎𝑑𝑗𝐴) = 𝑎𝑑𝑗𝐴. Also 𝑎𝑑𝑗𝐴 = (𝑎𝑑𝑗𝐴)(𝑎𝑑𝑗𝐴) ≥ (𝑎𝑑𝑗𝐴)𝐴 so that (𝑎𝑑𝑗𝐴)𝐴 = 𝑎𝑑𝑗𝐴. Thus we get 𝐴(𝑎𝑑𝑗𝐴) = (𝑎𝑑𝑗𝐴)𝐴 = 𝑎𝑑𝑗𝐴.

12

Example 4.13. [ ] ⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ 𝐿𝑒𝑡𝐴 = ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩ 𝑇 ℎ𝑒𝑛 𝐴11 = ⟨1, 1, 0⟩, 𝐴12 = ⟨0.6, 0.2, 0.3⟩ 𝐴21 = ⟨0.2, 0.3, 0.5⟩, 𝐴22 = ⟨1, 1, 0⟩ [ ][ ] ⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ 2 𝐴 = ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩ ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩ [ ] ⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ = ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩ 𝐴2 ≤ 𝐴i𝑠𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒. [ ] 𝐴11 𝐴21 𝑎𝑑𝑗𝐴 = 𝐴 𝐴22 [ 12 ⟨1, 1, 0⟩ 𝑎𝑑𝑗𝐴 = ⟨0.6, 0.2, 0.3⟩ [ ⟨1, 1, 0⟩ 𝐴(𝑎𝑑𝑗𝐴) = ⟨0.6, 0.2, 0.3⟩ [ ⟨1, 1, 0⟩ = ⟨0.6, 0.2, 0.3⟩ [ ⟨1, 1, 0⟩ (𝑎𝑑𝑗𝐴)𝐴 = ⟨0.6, 0.2, 0.3⟩ [ ⟨1, 1, 0⟩ = ⟨0.6, 0.2, 0.3⟩

⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩

]

⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩

][

⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩

]

⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩

][

⟨0.2, 0.3, 0.5⟩ ⟨1, 1, 0⟩

]

⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩

⟨1, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ ⟨0.6, 0.2, 0.3⟩ ⟨1, 1, 0⟩

]

]

(𝑎𝑑𝑗𝐴)𝐴 = 𝐴𝑎𝑑𝑗𝐴.

It is clear that the above example satisfies of the above Theorem.

Definition 4.14. An 𝑛 × 𝑛 FNSM 𝐴 is called circular if and only if (𝐴2 )𝑇 ≤ 𝐴, or more explicitly, ⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩⟨𝑎𝑇𝑘𝑖 , 𝑎𝐼𝑘𝑖 , 𝑎𝐹𝑘𝑖 ⟩ ≤ ⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩ for every 𝑘 = 1, 2, ...𝑛.

Theorem 4.15. For an 𝑛 × 𝑛 FNSM 𝐴 we have the following: (1)If 𝐴 is symmetric, then 𝑎𝑑𝑗𝐴 is symmetric, (2)If 𝐴 is transitive, then 𝑎𝑑𝑗𝐴 is transitive, 13

(3) If 𝐴 is circular, then 𝑎𝑑𝑗𝐴 is circular.

Proof. (1) Let 𝐵∑= 𝑎𝑑𝑗𝐴. ∏ Then 𝑇 𝐼 𝐹 ⟨𝑏𝑖𝑗 , 𝑏𝑖𝑗 , 𝑏𝑖𝑗 ⟩ = ⟨𝑎𝑇𝑡𝜋(𝑡) , 𝑎𝐼𝑡𝜋(𝑡) , 𝑎𝐹𝑡𝜋(𝑡) ⟩ 𝜋∈𝑆𝑛𝑗 𝑛𝑖 𝑡∈𝑛𝑗 ∑ ∏ 𝑇 ⟨𝑎𝜋(𝑡)𝑡 , 𝑎𝐼𝜋(𝑡)𝑡 , 𝑎𝐹𝜋(𝑡)𝑡 ⟩ = (⟨𝑏𝑇𝑖𝑗 , 𝑏𝐼𝑖𝑗 , 𝑏𝐹𝑖𝑗 ⟩).

𝜋∈𝑆𝑛𝑖 𝑛𝑗 𝑡∈𝑛𝑖

(since 𝐴 is symmetric).

(2)Let 𝐷 = 𝐴𝑖𝑗 . We can determine the elements of D in terms of the elements of 𝐴 as follows: ⎧  if ℎ < 𝑖, 𝑘 < 𝑗, ⟨𝑎𝑇ℎ𝑘 , 𝑎𝐼ℎ𝑘 , 𝑎𝐹ℎ𝑘 ⟩    ⎨⟨𝑎𝑇 𝐹 𝐼 if ℎ ≥ 𝑖, 𝑘 < 𝑗, (ℎ+1)𝑘 , 𝑎(ℎ+1)𝑘 , 𝑎(ℎ+1)𝑘 ⟩ ⟨𝑑𝑇ℎ𝑘 , 𝑑𝐼ℎ𝑘 , 𝑑𝑘ℎ𝑘 ⟩ = 𝐹 𝐼 𝑇  if ℎ < 𝑖, 𝑘 ≥ 𝑗, ⟨𝑎ℎ(𝑘+1) , 𝑎ℎ(𝑘+1) , 𝑎ℎ(𝑘+1) ⟩    ⎩⟨𝑎𝑇 𝐹 𝐼 (ℎ+1)(𝑘+1) , 𝑎(ℎ+1)(𝑘+1) , 𝑎(ℎ+1)(𝑘+1) ⟩ if ℎ ≥ 𝑖, 𝑘 ≥ 𝑗, where 𝐴𝑖𝑗 denotes the (𝑛 − 1) × (𝑛 − 1) FNSM obtained from 𝐴 by deleting the 𝑖 − 𝑡ℎ row and column 𝑗.

Now we show that 𝐴𝑠𝑡 𝐴𝑡𝑢 ≤ 𝐴𝑠𝑢 for every 𝑡 ∈ {1, 2, ...𝑛}. Let 𝑅 = 𝐴𝑠𝑡 , 𝐶 = 𝑇 𝐼 𝐹 𝐴𝑡𝑢 , 𝐹 = 𝐴𝑠𝑢 and 𝑊 = 𝐴𝑠𝑡 𝐴𝑡𝑢 . Note that 𝐴 is transitive. Then ⟨𝑤𝑖𝑗 , 𝑤𝑖𝑗 , 𝑤𝑖𝑗 ⟩ = 𝑛−1 ∑ 𝑇 𝐼 𝐹 𝑇 𝐼 𝐹 ⟨𝑟𝑖𝑘 , 𝑟𝑖𝑘 , 𝑟𝑖𝑘 ⟩⟨𝑐𝑘𝑗 , 𝑐𝑘𝑗 , 𝑐𝑘𝑗 ⟩ 𝑘=1 𝑛−1 ∑

=

= = = = = = ≤ = ≤

⟨𝑎𝑇𝑖𝑘 𝑎𝑇𝑘𝑗 , 𝑎𝐼𝑖𝑘 𝑎𝐼𝑘𝑗 , 𝑎𝐹𝑖𝑘 𝑎𝐹𝑘𝑗 ⟩ ≤ ⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩ 𝑘=1 ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 < 𝑠, 𝑘 < 𝑡, 𝑗 < 𝑢, 𝑛−1 ∑ 𝑇 𝑇 ⟨𝑎𝑖𝑘 𝑎𝑘(𝑗+1) , 𝑎𝐼𝑖𝑘 𝑎𝐼𝑘(𝑗+1) , 𝑎𝐹𝑖𝑘 𝑎𝐹𝑘(𝑗+1) ⟩ ≤ ⟨𝑎𝑇𝑖(𝑗+1) , 𝑎𝐼𝑖(𝑗+1) , 𝑎𝐹𝑖(𝑗+1) ⟩ 𝑘=1 ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 < 𝑠, 𝑘 < 𝑡, 𝑗 ≥ 𝑢, 𝑛−1 ∑ 𝑇 ⟨𝑎𝑖(𝑘+1) 𝑎𝑇(𝑘+1)𝑗 , 𝑎𝐼𝑖(𝑘+1) 𝑎𝐼(𝑘+1)𝑗 , 𝑎𝐹𝑖(𝑘+1) 𝑎𝐹(𝑘+1)𝑗 ⟩ ≤ ⟨𝑎𝑇𝑖𝑗 , 𝑎𝐼𝑖𝑗 , 𝑎𝐹𝑖𝑗 ⟩ 𝑘=1 ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 < 𝑠, 𝑘 ≥ 𝑡, 𝑗 < 𝑢, 𝑛−1 ∑ 𝑇 ⟨𝑎𝑖(𝑘+1) 𝑎𝑇(𝑘+1)(𝑗+1) , 𝑎𝐼𝑖(𝑘+1) 𝑎𝐼(𝑘+1)(𝑗+1) , 𝑎𝐹𝑖(𝑘+1) 𝑎𝐹(𝑘+1)(𝑗+1) ⟩ 𝑘=1 ⟨𝑎𝑇𝑖(𝑗+1) , 𝑎𝐼𝑖(𝑗+1) , 𝑎𝐹𝑖(𝑗+1) ⟩ = ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 < 𝑠, 𝑘 ≥ 𝑡, 𝑗 ≥ 𝑢, 𝑛−1 ∑ 𝑇 ⟨𝑎(𝑖+1)𝑘 𝑎𝑇𝑘𝑗 , 𝑎𝐼(𝑖+1)𝑘 𝑎𝐼𝑘𝑗 , 𝑎𝐹(𝑖+1)𝑘 𝑎𝐹𝑘𝑗 ⟩ 𝑘=1 ⟨𝑎𝑇(𝑖+1)𝑗 , 𝑎𝐼(𝑖+1)𝑗 , 𝑎𝐹(𝑖+1)𝑗 ⟩ = ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 ≥ 𝑠, 𝑘 < 𝑡, 𝑗 < 𝑢,

14

𝑛−1 ∑

⟨𝑎𝑇(𝑖+1)(𝑘+1) 𝑎𝑇(𝑘+1)𝑗 , 𝑎𝐼(𝑖+1)(𝑘+1) 𝑎𝐼(𝑘+1)𝑗 , 𝑎𝐹(𝑖+1)(𝑘+1) 𝑎𝐹(𝑘+1)𝑗 ⟩ 𝑘=1 ≤ ⟨𝑎𝑇(𝑖+1)𝑗 , 𝑎𝐼(𝑖+1)𝑗 , 𝑎𝐹(𝑖+1)𝑗 ⟩ = ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 ≥ 𝑠, 𝑘 ≥ 𝑡, 𝑗 < 𝑢, 𝑛−1 ∑ 𝑇 = ⟨𝑎(𝑖+1)(𝑘+1) 𝑎𝑇(𝑘+1)(𝑗+1) , 𝑎𝐼(𝑖+1)(𝑘+1) 𝑎𝐼(𝑘+1)(𝑗+1) , 𝑎𝐹(𝑖+1)(𝑘+1) 𝑎𝐹(𝑘+1)(𝑗+1) ⟩ 𝑘=1 ≤ ⟨𝑎𝑇(𝑖+1)(𝑗+1) , 𝑎𝐼(𝑖+1)(𝑗+1) , 𝑎𝐹(𝑖+1)(𝑗+1) ⟩ = ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 ≥ 𝑠, 𝑘 ≥ 𝑡, 𝑗 < 𝑢, 𝑛−1 ∑ 𝑇 = ⟨𝑎(𝑖+1)𝑘 𝑎𝑇𝑘(𝑗+1) , 𝑎𝐼(𝑖+1)𝑘 𝑎𝐼𝑘(𝑗+1) , 𝑎𝐹(𝑖+1)𝑘 𝑎𝐹𝑘(𝑗+1) ⟩ 𝑘=1 ≤ ⟨𝑎𝑇(𝑖+1)(𝑗+1) , 𝑎𝐼(𝑖+1)(𝑗+1) , 𝑎𝐹(𝑖+1)(𝑗+1) ⟩ = ⟨𝑓𝑖𝑗𝑇 , 𝑓𝑖𝑗𝐼 , 𝑓𝑖𝑗𝐹 ⟩ if 𝑖 ≥ 𝑠, 𝑘 < 𝑡, 𝑗 ≥ 𝑢, Thus 𝑤𝑖𝑗 𝑓𝑖𝑗 in every case and therefore ⟨𝑎𝑇𝑠𝑡 , 𝑎𝐼𝑠𝑡, 𝑎𝐹𝑠𝑡 ⟩ ∣𝐴𝑡𝑢 ∣ ≤ ∣𝐴𝑠𝑢 ∣ . for every 𝑡 ∈ {1, 2, ...𝑛} =

≤

By theorem (3.9) we get ∣𝐴𝑠𝑡 ∣ ∣𝐴𝑡𝑢 ∣ ≤ ∣𝐴𝑠𝑢 ∣ , This means that ⟨𝑏𝑇𝑡𝑠 , 𝑏𝐼𝑡𝑠 , 𝑏𝐹𝑡𝑠 ⟩⟨𝑏𝑇𝑢𝑡 , 𝑏𝐼𝑢𝑡 , 𝑏𝐹𝑢𝑡 ⟩ ≤ ⟨𝑏𝑇𝑢𝑠 , 𝑏𝐼𝑢𝑠 , 𝑏𝐹𝑢𝑠 ⟩, that is ⟨𝑏𝑇𝑢𝑡 , 𝑏𝐼𝑢𝑡 , 𝑏𝐹𝑢𝑡 ⟩⟨𝑏𝑇𝑡𝑠 , 𝑏𝐼𝑡𝑠 , 𝑏𝐹𝑡𝑠 ⟩ ≤ ⟨𝑏𝑇𝑢𝑠 , 𝑏𝐼𝑢𝑠 , 𝑏𝐹𝑢𝑠 ⟩, for every 𝑡 ∈ 1, 2, ..., 𝑛.

Hence 𝐵 = 𝑎𝑑𝑗𝐴 is transitive.

(3) Similarly, as in (2) we can show that 𝐴𝑠𝑡 𝐴𝑡𝑢 ≤ 𝐴𝑢𝑠 . For every 𝑡 ∈ 1, 2, ..., 𝑛 so that

∣𝐴𝑠𝑡 ∣ ∣𝐴𝑡𝑢 ∣ ≤ 𝐴𝑇𝑢𝑠 = ∣𝐴𝑢𝑠 ∣ . Thus ⟨𝑏𝑇𝑠𝑡 , 𝑏𝐼𝑠𝑡 , 𝑏𝐹𝑠𝑡 ⟩⟨𝑏𝑇𝑡𝑢 , 𝑏𝐼𝑡𝑢 , 𝑏𝐹𝑡𝑢 ⟩ ≤ ⟨𝑏𝑇𝑢𝑠 , 𝑏𝐼𝑢𝑠 , 𝑏𝐹𝑢𝑠 ⟩, and 𝐵 = 𝑎𝑑𝑗𝐴 is circular.

Corollary 4.16. If a FNSM 𝐴 is similarity then 𝐴𝑎𝑑𝑗𝐴 is also Similarity.

Theorem 4.17. For any 𝑛 × 𝑛 FNSM 𝐴 , the FNSM 𝐴𝑎𝑑𝑗𝐴 is transitive.

Proof.

L𝑒𝑡 𝐶 = 𝐴𝑎𝑑𝑗𝐴, tℎ𝑎𝑡𝑖𝑠 𝑛 ∑ ⟨𝑎𝑇𝑖𝑘 , 𝑎𝐼𝑖𝑘 , 𝑎𝐹𝑖𝑘 ⟩ ∣𝐴𝑗𝑘 ∣ 𝑐𝑖𝑗 = = 𝑐2𝑖𝑗 = =

𝑘=1 ⟨𝑎𝑇𝑖𝑓 , 𝑎𝐼𝑖𝑓 , 𝑎𝐹𝑖𝑓 ⟩ ∣𝐴𝑗𝑓 ∣ f𝑜𝑟𝑠𝑜𝑚𝑒𝑓 𝑛 ∑

∈ {1, 2, 3, ..𝑛}, a𝑛𝑑

𝑐𝑖𝑠 𝑐𝑠𝑗

𝑠=1 𝑛 ∑

𝑛 𝑛 ∑ ∑ 𝑇 𝐼 𝐹 ( (⟨𝑎𝑖𝑙 , 𝑎𝑖𝑙 , 𝑎𝑖𝑙 ⟩ ∣𝐴𝑠𝑙 ∣) ⟨𝑎𝑇𝑠𝑡 , 𝑎𝐼𝑠𝑡 , 𝑎𝐹𝑠𝑡 ⟩ ∣𝐴𝑗𝑡 ∣

𝑠=1 𝑙=1

𝑡=1

𝑛 ∑ = ⟨𝑎𝑇𝑖ℎ , 𝑎𝐼𝑖ℎ , 𝑎𝐹𝑖ℎ ⟩ ∣𝐴𝑠ℎ ∣ ⟨𝑎𝑇𝑠𝑢 , 𝑎𝐼𝑠𝑢 , 𝑎𝐹𝑠𝑢 ⟩ ∣𝐴𝑗𝑢 ∣

≤

𝑠=1 ⟨𝑎𝑇𝑖ℎ , 𝑎𝐼𝑖ℎ , 𝑎𝐹𝑖ℎ ⟩ ∣𝐴𝑗𝑢 ∣

15

â&#x2030;¤ â&#x;¨đ?&#x2018;&#x17D;đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; , đ?&#x2018;&#x17D;đ??źđ?&#x2018;&#x2014;đ?&#x2018;&#x201C; , đ?&#x2018;&#x17D;đ??šđ?&#x2018;&#x2014;đ?&#x2018;&#x201C; â&#x;Š â&#x2C6;Łđ??´đ?&#x2018;&#x2014;đ?&#x2018;&#x201C; â&#x2C6;Ł , fđ?&#x2018;&#x153;đ?&#x2018;&#x;đ?&#x2018; đ?&#x2018;&#x153;đ?&#x2018;&#x161;đ?&#x2018;&#x2019;â&#x201E;&#x17D;, đ?&#x2018;˘ â&#x2C6;&#x2C6; {1, 2, ..., đ?&#x2018;&#x203A;}.Tâ&#x201E;&#x17D;đ?&#x2018;˘đ?&#x2018; (đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)2 â&#x2030;¤ đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´. [

] â&#x;¨0.1, 0, 0.3â&#x;Š â&#x;¨0.5, 0.6, 0.7â&#x;Š Example 4.18. Letđ??´ be a FNSM . â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.6, 0.2, 0.3â&#x;Š [ ] â&#x;¨0.6, 0.2, 0.3â&#x;Š â&#x;¨0.5, 0.6, 0.7â&#x;Š Then (đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) = â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0, 0.3â&#x;Š [ ][ ] â&#x;¨0.1, 0, 0.3â&#x;Š â&#x;¨0.5, 0.6, 0.7â&#x;Š â&#x;¨0.6, 0.2, 0.3â&#x;Š â&#x;¨0.5, 0.6, 0.7â&#x;Š đ??´(đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) = â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨.6, .2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0, 0.3â&#x;Š [ ] â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0, 0.7â&#x;Š = â&#x;¨0.2, 0.2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š [ ][ ] â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0.0, 0.7â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0, 0.7â&#x;Š 2 (đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) = â&#x;¨0.2, 0.2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.2, 0.2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š [ ] â&#x;¨0.2, 0.5, 0.3â&#x;Š â&#x;¨0.1, 0, 0.7â&#x;Š (đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)2 = â&#x;¨0.2, 0.2, 0.3â&#x;Š â&#x;¨0.2, 0.5, 0.3â&#x;Š (đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´)2 â&#x2030;¤ (đ??´đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2014;đ??´) is also transitive.

We omit the proofs for type-II FNSM as the proofs are analogous to type-I FNSM. Conclusion: In this paper we have introduced determinant and adjoint of two types of FNSMs and discussed some of its properties.

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