Determinant Theory for Fuzzy Neutrosophic Soft Matrices by Ioan Degău

Progress in Nonlinear Dynamics and Chaos Vol. 4, No. 2, 2016, 85-102 ISSN: 2321 â&#x20AC;&#x201C; 9238 (online) Published on 30 November 2016 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/pindac.v4n2a5

Progress in

Determinant Theory for Fuzzy Neutrosophic Soft Matrices R.Uma1, P. Murugadas2 and S. Sriram3 1,2

Department of Mathematics, Annamalai University, Annamalainagar-608002, India Department of Mathematics, Mathematics wing, D.D.E, Annamalai University, Annamalainagar-608002, India Emil: 1uma83bala@gmail.com; 3ssm_3096@yahoo.co.in *Corresponding author. E-mail address: 2bodi_muruga@yahoo.com

Received 23 September 2016; accepted 17 October 2016 Abstract. We study determinant theory for fuzzy neutrosophic soft square matrices, its properties and also we prove that det ( Aadj ( A)) = det ( A) = det ( adj ( A) A). Keywords: Adjoint, determinant, fuzzy neutrosophic soft square matrix, determinant of fuzzy neutrosophic soft square matrix. AMS Mathematics Subject Classification (2010): 10M20 1. Introduction The complexity of problems in economics, engineering, environmental sciences and social sciences which cannot be solved by the well known methods of classical Mathematics pose a great difficulty in todayâ&#x20AC;&#x2122;s practical world (as various types of uncertainties are presented in these problems) . To handle situation like these, many tools have been suggested. Some of them are probability theory, fuzzy set theory [18], rough set theory [11], etc. The traditional fuzzy set is characterized by the membership value or the grade of membership value. Sometimes it may be very difficult to assign the membership value for fuzzy sets. Interval-valued fuzzy sets were proposed as a natural extension of fuzzy sets and the interval valued fuzzy sets were proposed independently by Zadeh [19] to ascertain the uncertainty of grade of membership value. In current scenario of practical problems in expert systems, belief 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 imprecise and doubtful environment. Neither the fuzzy sets nor the interval valued fuzzy sets is appropriate for such a situation. Intuitionistic fuzzy set initiated by Atanassov [3] 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 exist in belief system. The soft set theory, an utterly new theory for modeling ambiguity and uncertainties was first coined by Molodstov [9] in the year 1999. 85

R.Uma, P. Murugadas and S. Sriram Soft set theory research is carried out as a new trend and it shows much appreciable development well received by the users of the field. Fuzzy matrices play crucial role in Science and Technology. Sometimes the issues cannot be solved by classical matrix theory when they occur in an uncertain environment and this failure is inevitable. Thomason [14] initiated the fuzzy matrices to represent fuzzy relation in a system based on fuzzy set theory and discussed about the convergence of power of fuzzy matrix. In 1995, Smarandache introduced the concept of neutrosophy. In neutrosophic logic, each proposition is approximated to have the percentage of truth in a subset T, the percentage of indeterminancy in a subset I and the percentage of falsity in a subset F, so that this neutrosophic logic is called an extension of fuzzy logic. In fact this mathematical tool is used to handle problems like imprecision, indeterminancy and inconsistency of data etc. Maji et al. [5], initiated the concept of fuzzy soft set with some properties regarding fuzzy soft union, intersection, complement of fuzzy soft set. Moreover Maji et al. [6,10] extended soft sets to intuitionistic fuzzy soft sets and neutrosophic soft sets and the concept of neutrosophic set was introduced by Smarandache [12] which is a generalization of fuzzy logic and several related systems. Yang and Ji [17], introduced a matrix representation of fuzzy soft set and applied it in decision making problems. Bora et al. [8] introduced the intuitionistic fuzzy soft matrices and applied in the application of a Medical diagnosis. Sumathi and Arokiarani [13] introduced new operation on fuzzy neutrosophic soft matrices. Dhar et al. [7] have also defined neutrosophic fuzzy matrices and studied square neutrosophic fuzzy matrices. Uma et al. [15,16], introduced two types of fuzzy neutrosophic soft matrices and have discussed determinant and adjoint of fuzzy neutrosophic soft matrices. Kim et al. [4], introduced the concept of determinant theory for fuzzy matrices. In this paper, some elementary properties of determinant theory for fuzzy neutrosophic soft square matrices have been established and some theorems including det ( A( adjA)) = det ( A) = det ( adj ( A) A). where det ( A) denotes the determinant of A and adj ( A) denotes the adjoint matrix of A. 2. Preliminaries Definition 2.1. [12] A neutrosophic set A on the universe of discourse X is defined as A = {〈 x, TA ( x), I A ( x), FA ( x)〉, x ∈ X } , where T , I , F : X → ]− 0,1+ [ and − 0 ≤ TA ( x ) + I A ( x ) + FA ( x ) ≤ 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 scientific and Engineering problems it is difficult 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 ≤ TA ( x ) + I A ( x ) + FA ( x ) ≤ 3.

Determinant Theory for Fuzzy Neutrosophic Soft Matrices In short an element aɶ in the neutrosophic set A, can be written as aɶ = 〈 a , a I , a F 〉, where aT denotes degree of truth, a I denotes degree of indeterminacy, T

a F denotes degree of falsity such that 0 ≤ a T + a I + a F ≤ 3. Example 2.2. Assume that the universe of discourse X = {x1 , x2 , x3} , where x1 , x2 , and

x3 characterizes the quality, reliability, and the price of the objects. It may be further assumed that the values of {x1 , x2 , x3} 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 A is a Neutrosophic Set (NS) of X , such that A = {〈 x1 , 0.4, 0.5, 0.3〉 , 〈 x2 , 0.7, 0.2, 0.4〉 , 〈 x3 , 0.8, 0.3, 0.4〉}, where for x1 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. Definition 2.3. [9] Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set A, A ⊂ E . A pair (F,A) is called a soft set over U, where F is a mapping given by F : A → P (U ). Definition 2.4. [1] Let U be an initial universe set and E be a set of parameters. Consider a non empty set A, A ⊂ E . Let P (U ) denotes the set of all fuzzy neutrosophic sets of U . The collection ( F , A) is termed to be the Fuzzy Neutrosophic Soft Set (FNSS) over U, Where F is a mapping given by F : A → P (U ) . Hereafter we simply consider A as FNSS over U instead of ( F , A). Definition 2.5. [2] Let U = {c1 , c2 ,...cm } be the universal set and E be the set of

{

}

parameters given by E = e1 , e2 ,...en . Let A ⊆ E . A pair ( F , A) be a FNSS over U . Then the subset of U × E is defined by RA = {(u , e); e ∈ A, u ∈ FA (e)} which is called a relation form of ( FA , E ) . The membership function,

indeterminacy membership

function and non membership function are written by

TRA : U × E → [0,1] ,

I RA : U × E → [0,1] and FRA : U × E → [0,1] where TRA (u , e) ∈ [0,1], I RA (u , e) ∈ [0,1] and FRA (u , e ) ∈ [0,1] are the membership value, indeterminacy value and non

membership value respectively of u ∈ U for each e ∈ E . If [(Tij , I ij , Fij )] = [(Tij (ui , e j ), I ij (ui , e j ), Fij (ui , e j )] we define a matrix

R.Uma, P. Murugadas and S. Sriram

〈T12 , I12 , F12 〉 ⋯ 〈T1n , I1n , F1n 〉   〈T11 , I11 , F11 〉  〈T , I , F 〉 〈T , I , F 〉 ⋯ 〈T , I , F 〉  22 22 22 2n 2n 2n  〈Tij , I ij , Fij 〉  =  21 21 21 m×n   ⋮ ⋮ ⋮    〈Tm1 , I m1 , Fm1 〉 〈Tm 2 , I m 2 , Fm 2 〉 ⋯ 〈Tmn , I mn , Fmn 〉  This is called an m × n FNSM of the FNSS (FA, E) over U. Definition 2.6. [15] Let U = {c1 , c2 ...cm } be the universal set and E be the set of

parameters given by E = {e1 , e2 ,...en } . Let A ⊆ E . A pair ( F , A) be a fuzzy neutrosophic soft set. Then fuzzy neutrosophic soft set ( F , A) in a matrix form as

Am×n = ( aij ) m×n or A = ( aij ), i = 1, 2,...m, j = 1, 2,...n where

(T (ci , e j ), I (ci , e j ), F (ci , e j )) if e j ∈ A (aij ) =  if e j ∉ A 〈0, 0,1〉 where T j (ci ) represent the membership of ci , I j (ci ) represent the indeterminacy of ci and F j (ci ) represent the non-membership of ci in the FNSS

( F , A) . If we replace the

identity element 〈 0, 0,1〉 by 〈 0,1,1〉 in the above form we get FNSM of type-II. Let Fm×n denotes FNSM of order m × n and Fn denotes FNSM of order n × n . Definition 2.7. [15] [Type-I] Let A = ( aijT , aijI , aijF ), B = ( bijT , bijI , bijF ) ∈ Fm×n

the component wise addition and

component wise multiplication is defined as 1. A ⊕ B = ( sup aijT , bijT , sup aijI , bijI , inf aijF , bijF ).

{

}

{

}

{

}

2. A ⊙ B = (inf {aijT , bijT }, inf {aijI , bijI }, sup{aijF , bijF }). Definition 2.8. [15] Let A ∈ Fm×n , B ∈ Fn× p , the composition of A and B is defined as

 n A B =  ∑ (aikT ∧ bkjT ),  k =1

∑ (aikI ∧ bkjI ), k =1

∏ (a

F ik

k =1

 ∨ bkjF )  

equivalently we can write the same as

( ∨ (a ∧ b ), ∨ (a ∧ b ), ∧ (a ∨ b ) ) n

k =1

The product A B is defined 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. We shall use AB instead of A B. Definition 2.9.[15][Type-II] Let

A = (〈 aijT , aijI , aijF 〉 ), B = (〈bijT , bijI , bijF 〉 ) ∈ Fm×n , the

component wise addition and component wise multiplication is defined as 88

Determinant Theory for Fuzzy Neutrosophic Soft Matrices

{

}

{

}

{

}

A ⊕ B = (〈 sup aijT , bijT , inf aijI , bijI , inf aijF , bijF 〉 ).

A ⊙ B = (〈inf {a , b }, sup{a , b }, sup{a , b }〉 ). T ij

T ij

I ij

F ij

Analogous to FNSM of type-I, we can define FNSM of type -II in the following way T I F Definition 2.10. [15] Let A = (〈 aij , aij , aij 〉 ) = (aij ) ∈ Fm×n and

B = (〈bijT , bijI , bijF 〉 ) = (bij ) ∈ Fn× p the product of A and B is defined as  n A ∗ B =  ∑ aikT ∧ bkjT ,  k =1

∏ k =1

aikI ∨ bkjI ,

∏ k =1

 aikF ∨ bkjF  

equivalently we can write the same as n n  n  =  ∨ aikT ∧ bkjT , ∧ aikI ∧ bkjI , ∧ aikF ∨ bkjF  . = 1 = 1 = 1 k k k   The product A ∗ B is defined 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. Definition 2.11. [16] The determinant A of n × n FNSM A = (〈 aijT , aijI .aijF 〉 ) is defined as follows

A = 〈 ∨ a1Tσ (1) ∧ ... ∧ anTσ ( n ) , ∨ a1Iσ (1) ∧ ... ∧ anIσ ( n ) , ∧ a1Fσ (1) ∨ ... ∨ anFσ ( n ) 〉 σ ∈Sn

σ ∈Sn

where Sn denotes the symmetric group of all permutations of the indices (1, 2,...n). Definition 2.12. [16] The adjoint of an n × n FNSM A denoted by adj A, is defined as follows bij = A ji is the determinant of the ( n − 1) × ( n − 1) FNSM formed by deleting row j and column i from A and B = adjA . Remark 2.13. We can write the element bij of adjA = B = (bij ) as follows:

bij =

∑∏ π ∈Sn j ni t∈n j

atTπ (t ) , atIπ (t ) , atFπ (t ) Where n j = {1, 2,3.....n} \{ j} and Sn j ni is the set of

all permutation of set n j over the set ni . 3. Properties of the fuzzy neutrosophic soft square matrices (FNSSM) 1. The value of the determinant remains unchanged when any two rows or columns are interchanged. 2. The values of the determinant of FNSSM remain unchanged when rows and columns are interchanged. 3. If A and B be two FNSSMs then the following property will hold

det ( AB ) ≠ detA detB.

4. If the elements of any row (or column) of a determinant are added to the corresponding elements of another row (or column), the value of the determinant thus obtained is equal to the value of the original determinant. 89

R.Uma, P. Murugadas and S. Sriram T I F Theorem 3.1. Let A = (aij , aij , aij ) ∈ FNSSMn. T Let Ak = (〈 akT1 , akI1 , akF1 〉 , 〈 akT2 , akI 2 , akF2 〉 ,..., 〈 akn , aknI , aknF 〉 ) be the k -th row of A. We

assume that a1 = 〈 akiT , akiI , akiF 〉 for all i ∈ 1, 2,..., n and a pq ≥ a1 for all p, q ∈ 1, 2,..., n. Then det ( A) = a1. Theorem 3.2. Let A ∈ FNSSM n then (i) det ( A) = A =

∑ 〈a t =1

T it

, aitI , aitF 〉 Ait , i ∈ {1, 2,..., n}.

〈 a1Te , a1Ie , a1Fe 〉

〈 a1Tf , a1I f , a1Ff 〉

1 A , a , a 〉 〈a , a , a 〉  e e< f taken over all e and f in {1,2,...,n} such that e < f .

(ii) det ( A) =

∑ 〈a

T 2e

I 2e

F 2e

T 2f

I 2f

F 2f

2  , where the summation is f

T I F Definition 3.3. Let A = (aij , aij , aij ) n ∈ FNSSM n , and let B be a matrix from A by

striking out e1 , row e2 ,..., row ek and column g1 , column g 2 , ..., column g k . we define

 e1 A  g1

...ek   = det ( H ). ...g k 

e2 g2

Theorem 3.4.

 〈 a1T g1 , a1I g1 , a1F g1 〉  T I F  〈 a2 g1 , a2 g 2 , a2 g 2 〉 det ( A) = ∑ det   g1 < g 2 <...< g k ⋮ ⋮ ⋮  T  〈a g , a I g , a k g 〉  k 1 k 1 1 1

1 A  g1

...〈 a1T g k , a1T g k , a1T g k 〉   ...〈 a2T g k , a2I g k , a2F g k 〉     T I F ...〈 ak g k , ak g k , ak g k 〉 

...k   where the summation is taken over all g1 , g 2 ,..., g k ∈ {1, 2,...n}, ...g k 

2 g2

such that g1 < g 2 < ... < g k . Proof: Let S ( g1 , g 2 ,..., g k ) = {σ :{1, 2,..., k } → {g1 , g 2 ,..., g k } σ is a bijection}. Then

det ( A) =

∑ 〈a σ σ ∈Sn

∑

g1 < g 2 <...< g k

T 1 (1)

, a1Iσ (1) , a1Fσ (1) 〉...〈 anTσ ( n ) , anIσ ( n ) , anFσ ( n ) 〉

( ∑σ {1,2,...,k }∈S ( g1 , g2 ,..., gk )〈 a1Tσ (1) , a1Iσ (1) , a1Fσ (1) 〉...〈 anTσ ( n ) , anIσ ( n ) , anFσ ( n ) 〉 )

Determinant Theory for Fuzzy Neutrosophic Soft Matrices

∑

g1 < g 2 <...< g k

( ∑σ ' {1,2,...,k }∈S ( g , g 1

2 ,..., g k

)

)(〈 a1Tσ ' (1) , a1Iσ ' (1) , a1Fσ ' (1) 〉...〈 anTσ ' ( n ) , anIσ ' ( n ) , anFσ , ( n ) 〉 )

 1 2 ... k  A   g1 g2 ... gk 

〈a1T g1, a1I g1, a1F g1〉 〈a1T g2, a1I g2, a1F g2〉  T I F T I F 〈a2 g1, a2 g1, a2 g1〉 〈a2 g2, a2 g2, a2 g2〉  ... ...  T I F T I F 〈ak g1, ak g1, ak g1〉 〈ak g2, ak g2, ak g2〉

∑

det

g1 < g 2 <...< g k

... 〈a1T gk , a1I gk , a1F gk 〉   ... 〈a2T gk , a2I gk , a2F gk 〉   ... ...  ... 〈akT gk , akI gk , akF gk 〉 

 1 2 ... k  A   g1 g2 ... gk 

Hence the pooof. Lemma 3.5.

 〈 aT , a I , a F 〉

〈bT , b I , b F 〉  be a FNSSM. T I F T I F   〈c , c , c 〉 〈 d , d , d 〉   〈 a T , a I , a F 〉 〈 bT , b I , b F 〉   〈 cT , c I , c F 〉 〈 d T , d I , d F 〉  Then det  T I F det  T I F T I F  T I F   〈 a , a , a 〉 〈b , b , b 〉   〈c , c , c 〉 〈 d , d , d 〉 

Let A = 

〈 aT , a I , a F 〉

〈bT , b I , b F 〉 〈cT , c I , c F 〉

〈d T , d I , d F 〉

〈 aT , a I , a F 〉 〈bT , b I , b F 〉 〈cT , c I , c F 〉 〈 d T , d I , d F 〉

≤ det ( A)

Proof:

 〈 aT , a I , a F 〉 〈bT , b I , b F 〉  det  T I F T I F   〈 a , a , a 〉 〈b , b , b 〉  = 〈 aT , a I , a F 〉〈bT , b I , b F 〉〈cT , c I , c F 〉〈 d T , d I , d F 〉

see

that

 〈 cT , c I , c F 〉 〈 d T , d I , d F 〉  det  T I F T I F   〈c , c , c 〉 〈 d , d , d 〉 

≤ 〈 aT , a I , a F 〉〈 d T , d I , d F 〉 + 〈bT , b I , b F 〉〈cT , c I , c F 〉 = det ( A). Notation: Let A ∈ FNSSM n .Let A(e ⇒ f ) be the matrix obtained from A by replacing row f of A by row e of A. Theorem 3.6. Let A ∈ FNSSM n . Then

(i )det ( A(2 ⇒ 1))det ( A(1 ⇒ 2)) ≤ det ( A). (ii )det ( A(2 ⇒ 1))det ( A(3 ⇒ 2)) ≤ det ( A). (iii )det ( A(q ⇒ p )))det ( A( p ⇒ k )) ≤ det ( A). Proof: To prove

R.Uma, P. Murugadas and S. Sriram

(i )det ( A(2 ⇒ 1))det ( A(1 ⇒ 2)) =

〈a , a , a 〉 〈a , a , a 〉 〈a , a , a 〉 〈a , a , a 〉 T 11 T 11

I 11 I 11

F 11 F 11

T 12 T 12

I 12 I 12

F 12 F 12

〈 a1Te , a1Ie , a1Fe 〉 〈 a1Tf , a1I f , a1Ff 〉 1 2  A  + ... + T I F 〈 a1e , a1e , a1e 〉 〈 a1Tf , a1I f , a1Ff 〉 1 2  e< f

1 2  A  + ... + e f  〈 a1Tn −1 , a1In −1 , a1Fn −1 〉 〈 a1Tn , a1In , a1Fn 〉

〈 a1Tn −1 , a1In −1 , a1Fn −1 〉 〈 a1Tn , a1In , a1Fn 〉 1 2    + ... + 1 2  〈 a2Te , a2I e , a2Fe 〉 〈 a2T f , a2I f , a2Ff 〉 〈 a1Te , a1Ie , a1Fe 〉 〈 a1Tf , a1I f , a1Ff 〉

〈a 〈a

,a ,a

1  g

2  h

2  1 .  n −1 n 

A

1 e

A e< f

T I F T I F 〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉 T I F T I F 〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉

2  + ... + f

2 〉 〈a , a , a 〉  1  A 〉 〈a , a , a 〉  n − 1 n  〈 a1Te , a1Ie , a1Fe 〉 〈 a1Tf , a1I f , a1Ff 〉  1 2  〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉 A A =∑ T I F ∑ 〈 a1Tf , a1I f , a1Ff 〉  e f  g < h 〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉 e < f 〈 a1e , a1e , a1e 〉 T 2 n −1 T 2 n −1

≤∑ e< f g <h

I 2 n −1 I 2 n −1

F 2 n −1 F 2 n −1

,a ,a

〈 a1Te , a1Ie , a1Fe 〉

T 2n T 2n

I 2n I 2n

F 2n F 2n

〈 a1Tf , a1I f , a1Ff 〉

〈a , a , a 〉 〈a , a , a 〉 T 2g

I 2g

F 2g

T 2h

I 2h

F 2h

1 e

A

e g

We now introduce symbols J 1 , J 2 , A 

2 1 A f  g

2  h

(by Lemma 3.5)

f  and J . Define h

f  〈 a1Te , a1Ie , a1Fe 〉 〈 a1Tf , a1I f , a1Ff 〉 1 2   1 A = T I F A T I F h  〈 a2 g , a2 g , a2 g 〉 〈 a2 h , a2 h , a2 h 〉 e f   g e f  e f  J1 = ∑ J   = ∑J  , h  e< f  e f  ( e , f ) =( g , h )  g e f  J2 = ∑ J   and J = J1 + J 2 . Then we see that h (e, f )≠ ( g ,h )  g e J g

1 2  〈 a11T , a11I , a11F 〉 〈 a12T , a12I , a12F 〉 1 2  A J = T I F , T I F 1 2  〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉 1 2  J 1 = det ( A) by Theorem 3.2(ii) and det ( A(2 ⇒))det ( A(1 ⇒ 2)) ≤ J = J 1 + J 2 = det ( A) + J 2 . 92

2  h

Determinant Theory for Fuzzy Neutrosophic Soft Matrices We show that J 2 ≤ det ( A). There are two cases to be considered.

1 2   , a term of J 2 . 1 3  1 2  1 2  T T I F Let a1 = 〈 a11 , a11I , a11F 〉〈 a23 , a23 , a23 〉 A A  and 1 2  1 3  1 2  1 2  T I F a2 = (〈 a12T , a12I , a12F 〉〈 a21 , a21 , a21 〉 A A  1 2  1 3  Then a = a1 + a2 ,

Case 1. We consider a = J 

〈 a11T , a11I , a11F 〉

〈 a13T , a13I , a13F 〉

1 3   ≤ det ( A), 〈 a , a , a 〉 〈 a , a , a 〉 1 3  〈 aT , a I , a F 〉 〈 a12T , a12I , a12F 〉 1 2  a2 ≤ T11 11I 11F A  ≤ det ( A), T I F , a22 , a22 〈 a21 , a21 , a21 〉 〈 a22 〉 1 2  a1 ≤

T 21

I 21

F 21

T 23

I 23

F 23

A

1 2   ≤ det ( A), 1 3  2  1 Case 2. We take J    n −1 n 

and J 

2 1 2   1 A   and 1 2   n − 1 n  2 1 2   1 b2 = 〈 a12T , a12I , a12F 〉〈 a2Tn −1 , a2I n −1 , a2Fn −1 〉 A  A  . 1 2   n − 1 n  2  1 Then J   = b1 + b2 . To show that b1 ≤ det ( A) and b2 = det ( A) , we  n −1 n  2 1 2   1 observe all coordinates of the elements aij involved in A   and A  . 1 2   n −1 n  The coordinates of the elements aij involved in these determinants are all coordinates of T Let b1 = 〈 a11 , a11I , a11F 〉〈 a2Tn , a2I n , a2Fn 〉 A 

the elements of the k − th row

Ak of A, for k ≥ 3. Therefore, if we let

b = a3n −1a4 n − 2 ...ak + 2 n − k ...ann −2 , then we see that T b1 ≤ (〈 a11 , a11I , a11F 〉〈 a2Tn , a2I n , a2Fn 〉 )c ≤ det ( A). For b2 , let c = a3n a4 n −2 a5 n−3 ...an −13a2 n −1 ,

then

e J g

see

that

T b2 ≤ (〈 a12 , a12I , a12F 〉〈 a2Tn −1 , a2I n −1 , a2Fn −1 〉 )c ≤ det ( A).

For

f ,we apply either the case 1 or the case 2 and we can deduce that  h ( e , f ) ≠ ( g , h ) 93

any

R.Uma, P. Murugadas and S. Sriram

e J  g

f ≤ det ( A).  h ( e , f ) ≠ ( g , h )

Thus (i) holds. (ii). First we consider T T 〈b11T , b11I , b11F 〉 〈b12T , b12I , b12F 〉 〈b21 , b21I , b21F 〉 〈b22 , b22I , b22F 〉 T T T T 〈 a31 , a31I , a31F 〉 〈 a32 , a32I , a32F 〉 〈b21 , b21I , b21F 〉 〈b22 , b22I , b22F 〉

≤

T 〈b21 , b21I , b21F 〉

T 〈b32 , b32I , b32F 〉

T T 〈 a31 , a31I , a31F 〉 〈 a32 , a32I , a32F 〉

We introduce a symbol

h  〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉  2 A = f  〈 a3Te , a3Ie , a3Fe 〉 〈 a3T f , a3I f , a3Ff 〉  g

g K e

3 A h

2  e

3 . f

Then we can see that

det ( A(2 ⇒ 1))det ( A(3 ⇒ 2)) =

∑

e< f g <h

〈 a1Te , a1Ie , a1Fe 〉

〈 a1Tf , a1I f , a1Ff 〉

2 A T I F T I F 〈 a3e , a3e , a3e 〉 〈 a3 f , a3 f , a3 f 〉  e

3  〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉 A  f  〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉

 2 3    g h 〈 a2Tg , a2I g , a2Fg 〉 〈 a2Th , a2I h , a2Fh 〉  2 3 A ≤∑ T I F A T I F 〈 a3 f , a3 f , a3 f 〉  g h  g < h 〈 a3 e , a3e , a3 e 〉 e< f

g

∑ A e g <h e< f



2  e

3  f

h  f

h g h + ∑ K  . f  ( g ,h) ≠( e, f )  e f  g h Next we prove that K  ≤ det ( A).   e f ( g ,h ) ≠ ( e , f )

g K ( g ,h)=(e, f ) e

∑

For this we consider two cases.

1 2   . We see that 1 3 

Case 1. We take K 

1 2   2 3 T I F T I F T I F T I F K  =( 〈 a21 , a21 , a21 〉〈 a33 , a33 , a33 〉 + 〈 a22 , a22 , a22 〉〈 a31 , a31 , a31 〉 ) A  A 1 3  1 2  2 3    1 3 94

Determinant Theory for Fuzzy Neutrosophic Soft Matrices

 2 3  2 3 T I F T I F  A  +( 〈 a22 , a22 , a22 〉〈 a31 , a31 , a31 〉 ) A  1 2  1 3

T I F T = (〈 a21 , a21 , a21 〉〈 a33 , a33I , a33F 〉 ) A 

 2 3  2 3   A   1 2  1 3 I F T I F T I F T I F 〈 aT , a21 , a21 〉 〈 a23 , a23 , a23 〉  2 3  〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉  2 3 ≤ 21 A + A    T I F T I F T I F T I F 〈 a31 , a31 , a31 〉 〈 a33 , a33 , a33 〉  1 3  〈 a31 , a31 , a31 〉 〈 a32 , a32 , a32 〉  1 2  ≤ det ( A) + det ( A) = det ( A). Case 2. We consider

 n −1 n  2 T I F T I F K  = 〈 a2 n −1 , a2 n −1 , a2 n −1 〉〈 a32 , a32 , a32 〉 A  2  1 1 2 T 〈 a2Tn , a2I n , a2Fn 〉〈 a31 , a31I , a31F 〉 A  1

3  2 3  A + 2   n −1 n  3  2 3  A . 2   n −1 n  3  2 3  2 Considering the coordinates of the elements aij involved in A  A  , we  1 2   n −1 n  claim that

3  2 3  2 T (〈 a2Tn −1 , a2I n −1 , a2Fn −1 〉〈 a32 , a32I , a32F 〉 ) A   A  ≤ det ( A)  1 2   n −1 n  and

3  2 3  2 T (〈 a2Tn , a2I n , a2Fn 〉〈 a31 , a31I , a31F 〉 A   A  ≤ det ( A).  1 2   n −1 n  Similarly we can prove (iii). Theorem 3.7. Let A = (aijT , aijI , aijF ) n , B = (bijT , bijI ,ijF ) n , (cijT , cijI , cijF ) n ∈ FNSSM n . Then 1.

〈 aiiT , aiiI , aiiF 〉 ≥ 〈 aikT , aikI , aikF 〉 ( k = 1, 2,..., n)

for

all

1 ≤ i ≤ n,

det ( A) = 〈 a , a , a 〉〈 a , a , a 〉...〈 a , a , a 〉. T 11

T 11

T 22

T nn

 A C  det ( A) det ( B ) where O = (〈 0, 0,1〉 ) n ∈ FNSSM n O B 3. det ( AAT ) ≥ det ( A). 2. det 

T I F T I F 4. If 〈 aij , aij , aij 〉 ≥ 〈bij , bij , bij 〉 for all i, j, then det ( A) = det ( B ).

Proof: 1. We have T I F T I F 〈 a11T , a11I , a11F 〉〈 a22 , a22 , a22 〉...〈 ann , ann , ann 〉 ≥ 〈 a1Tσ (1) , a2Iσ (2) , anFσ ( n ) 〉 for every σ ∈ S n ,

since 〈 aiiT , aiiI , aiiF 〉 ≤ 〈 aikT , aikI , aikF 〉 ( k = 1, 2,..., n) for all 1 ≤ i ≤ n. 95

then

R.Uma, P. Murugadas and S. Sriram

∑ 〈a σ σ

Hence det ( A) =

T 1 (1)

∈Sn

, a1Iσ (1) , a1Fσ (1) 〉〈 a2Tσ (2) , a2Iσ (2) , a2Fσ (2) 〉...〈 anTσ ( n ) , anIσ ( n ) , anFσ ( n ) 〉

I F T I F = 〈 a , a , a 〉〈 a , a22 , a22 〉...〈 ann , ann , ann 〉 T 11

I 11

F 11

T 22

 A C T I F  = (〈 dij , dij , dij 〉 )2 n . O B  

2. Let  Then

 A C T I F T I F det   = ∑ 〈 d1σ (1) , d1σ (1) , d1σ (1) 〉...〈 d 2 nσ (2 n ) , d 2 nσ (2 n ) , d 2 nσ (2 n ) 〉 O B   σ ∈S2 n 〈 d1Tσ (1) , d1Iσ (1) , d1Fσ (1) 〉...〈 d 2Tnσ (2 n ) , d 2I nσ (2 n ) , d 2Fnσ (2 n ) 〉 + = ∑ σ ∈S2 n ,σ ( i ) ≤ n (if i ≤ n )

∑σ

〈 d1Tσ (1) , d1Iσ (1) , d1Fσ (1) 〉...〈 d 2Tnσ (2 n ) , d 2I nσ (2 n ) , d 2Fnσ (2 n ) 〉

∑ σ

〈 d1Tσ (1) , d1Iσ (1) , d1Fσ (1) 〉...〈 d 2Tnσ (2 n ) , d 2I nσ (2 n ) , d 2Fnσ (2 n ) 〉 + 0

σ ∈S2 n ,∃k > n ,if ( k ) ≤ n

σ ∈S2 n , ( i ) ≤ n ( ifi ≤ n )

∑ 〈d σ σ

, d1Iσ '(1) , d1Fσ '(1) 〉... 〈 d nTσ '( n ) , d nIσ '( n ) , d nFσ '( n ) 〉 det ( B)

∑ 〈d σ σ

, d1Iσ (1) , d1Fσ (1) 〉...〈 d nTσ ( n ) , d nIσ ( n ) , d nFσ ( n ) 〉 )det ( B )

'∈Sn

∈Sn

T 1 '(1)

T 1 (1)

= det ( A) det ( B ). T T I F 3. Let AA = (〈 gij , g ij , gij 〉 )n , where 〈 g ijT , g ijI , g ijF 〉 =

We have, for every σ ∈ Sn

∑ 〈a k =1

T ik

, aikI , aikF 〉〈 akjT , akjI , akjF 〉.

T I F T I F 〈 g11T , g11I , g11F 〉〈 g 22 , g 22 , g 22 〉...〈 g nn , g nn , g nn 〉

= (

∑ 〈 a1Tk , a1Ik , a1Fk 〉 )...(∑ 〈ankT , ankI , ankF 〉 ) k =1 T 1σ (1)

≥ 〈a

I 1σ (1)

F 1σ (1)

k =1 T nσ ( n )

〉...〈 a

, anIσ ( n ) , anFσ ( n ) 〉

T T T I F Hence det ( AAT ) ≥ 〈 g11 , g11I , g11F 〉〈 g 22 , f 22I , f 22F 〉...〈 g nn , g nn , g nn 〉

≥

∑ 〈a σ σ ∈Sn

T 1 (1)

, a1Iσ (1) , a1Fσ (1) 〉...〈 anTσ ( n ) , anIσ ( n ) , anFσ ( n ) 〉

= det ( A). Theorem 3.8. Let A = ( aij ) be a FNSSM. Then we have the following

det ( Aadj ( A)) = det ( A) = det ( adj ( A) A). Proof: We prove that det ( Aadj ( A)) = det ( A) . We first consider n=2.

Determinant Theory for Fuzzy Neutrosophic Soft Matrices

 〈 a11T , a11I , a11F 〉 〈 a12T , a12I , a12F 〉  . Let A =  T I F T I F   〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉  Then we see that

 〈 aT , a I , a F 〉 〈 a12T , a12I , a12F 〉  adj ( A) =  22T 22I 22 . F T I F   〈 a21 , a21 , a21 〉 〈 a11 , a11 , a11 〉  det ( A) 〈 a11T , a11I , a11F 〉〈 a12T , a12I , a12F 〉 det ( Aadj ( A)) = T I F T I F 〈 a21 , a21 , a21 〉〈 a22 , a22 , a22 〉 det ( A) T T I F T I F = det ( A) + (〈 a11 , a11I , a11F 〉〈 a12T , a12I , a12F 〉〈 a21 , a21 , a21 〉〈 a22 , a22 , a22 〉)

≤ det ( A). Next consider n > 2. We can see that

 ∑ 〈 a1Tt , a1It , a1Ft 〉 A1t  T I F  ∑ 〈 a2t , a2t , a2t 〉 A1t  ...  T  ∑ 〈a , aI , a F 〉A nt nt 3t 1t 

Aadj ( A) =

( ∑〈a , a , a 〉A ) , det ( Aadj( A)) = ∑ (∑ 〈a , a , a

T it

I it

F it

∑ 〈a ∑ 〈a

T 1t T 2t

, a1It , a1Ft 〉 A2t , a2I t , a2Ft 〉 A2t

... ∑ 〈a , antI , antF 〉 A2t T nt

∑ 〈a ∑ 〈a

, a1It , a1Ft 〉 Ant   , a2I t , a2Ft 〉 Ant   ... ...  T I F ... ∑ 〈 ant , ant , ant 〉 Ant  ... ...

T 1t T 2t

π ∈Sn

T 1t

I 1t

F 1t

〉 Aπ (1)t )(∑ 〈a2Tt , a2I t , a2Ft 〉 Aπ (2)t )...(∑ 〈antT , antI , antF 〉 Aπ ( n)t ).

Clearly any diagonal entry of the matrix Aadj ( A) is equal to det ( A). We prove the result in the following way. (1) Let us define

Tπ = (∑ 〈 a1Tt , a1It , a1Ft 〉 Aπ (1)t )(∑ 〈 a2Tt , a2I t , a2Ft 〉 Aπ (2)t )...(∑ 〈 antT , antI , antF 〉 Aπ ( n )t ),

for π ∈ S n . Let e be the identity of the group S n . If π = e, then Tπ = det ( A). Suppose that there exists k ∈ {1, 2,..., n} such that π (k ) = k . Then we see that

∑ 〈a

a a 〉 Aπ ( k )t = ∑ 〈 aktT aktI aktF 〉 Akt = det ( A) and J π = (∑ 〈 a1Tt , a1It , a1Ft 〉 Aπ (1)t )(∑ 〈 a2Tt , a2I t , a2Ft 〉 Aπ (2)t )...det ( A)...(∑ 〈 antT , antI , antF 〉 Aπ ( n )t ) ≤ det ( A). (2) Let π be a permutation in S n . Assume that π ( k ) ≠ k for all k ∈ {1, 2,...n}. We know that every permutation π can be written as a product of disjoint cycles π i and let π = π 1π 2 ...π k . We further assume that π 1 = (1 2), a transposition. T I F kt kt kt

Then J π has two factors, (

∑ 〈a

T 1t

, a1It , a1Ft 〉 Aπ (1)t ) and

(∑ 〈 a2Tt , a2I t , a2Ft 〉 Aπ (2)t ), and from these we see that

R.Uma, P. Murugadas and S. Sriram

(∑ 〈 a1Tt , a1It , a1Ft 〉 Aπ (1) t )(∑ 〈 a2Tt , a2I t , a2Ft 〉 Aπ (2)t ) = (∑ 〈 a1Tt , a1It , a1Ft 〉 A2t )( ∑〈 a2Tt , a2I t , a2Ft 〉 A1t ) = det ( A(2 ⇒ 1)) det ( A(1 ⇒ 2)) ≤ det ( A) (by theorem3. 2(i)) (3) If π = π 1π 2 ...π k and π 1 ( s, t ), then we can prove that J π ≤ det ( A) by an argument

used in(2). Consider J π for π = π1π 2 ...π k . If

π1 = (k , e, f ,...), then we see that

J π = (∑ 〈 aktT , aktI , aktF 〉 Aπ ( k ) t ) )(∑ 〈 aetT , aetI , aetF 〉 Aπ ( e ) t ) )... = (∑ 〈 aktT , aktI , aktF 〉 Aet )( ∑〈 aetT , aetI , aetF 〉 Aft )... = det ( A(e ⇒ k ))det ( A( f ⇒ e))... From Theorem 3.6(iii), we obtain that det ( A(e ⇒ k )) det ( A( f ⇒ e)) ≤ det ( A) and consequently that J π ≤ det ( A). This proves that det ( Aadj ( A)) = det ( A). Similarly, we can prove that det ( adj ( A) A) = det ( A). Hence the proof. Theorem 3.9. Let A, B ∈ FNSSM n . Then

(1) det ( AB ) ≥ det ( A)det ( B ). (2) det ( AB ) ≤ det ( A + B ), where A + B = ( sup{aijT , bijT }, sup{aijI , bijI }, inf {aijF , bijF }) Proof. n

det ( AB ) = det ((∑ aikT ∧ bkjT , ∑ aikI ∧ bkjI , ∏ aikF ∨ bkjF )) k =1

∑ {[∑ a σ ∈S n

k =1

T nk

T 1k

∧ bkTσ (1) , ∑ a1Ik ∧ bkIσ (1) , ∏ aikF ∨ bkFσ (1) )],⋯ , k =1

k =1

∧ bkTσ ( n ) , ∑ ankI ∧ bkIσ ( n ) , ∏ (ankF ∨ bkFσ ( n ) ]} k =1

∑( ∑ σ ∑

k =1

(a ∧ a ... ∧ a

∈Sn k1 , k 2 ...k n

(

k =1

[∑ a k =1

k =1

T 1k1

T 2 k2

T nkn

∧ bkT1σ (1) ∧ bkT2σ (2) ... ∧ bkTnσ ( n ) )

(a1Ik1 ∧ a2I k2 ... ∧ ankI n ∧ bkI1σ (1) ∧ bkI2σ ( 2 ) ... ∧ bkInσ ( n ) )

k1 , k2 ... kn

∏

(a1Fk1 ∨ a2Fk2 ∨ ...ankF n ∨ bkF1σ (1) ∨ bkF2σ ( 2 ) ∨ ...bkFnσ ( n ) )

k1 ,k2 ...kn

≥

∑

{k1 ,k2 ...kn }∈Sn

T 〈 a1Tk1 , a1Ik1 , a1Fk1 〉...〈 ank , ankI n , ankF n 〉 n

Determinant Theory for Fuzzy Neutrosophic Soft Matrices

∑

〈bkT1σ (1) , bkI1σ (1) , bkF1σ (1) 〉...〈bkTnσ (n) , bkInσ (n) , bkFnσ (n) 〉) =

σ∈Sn

∑

(

T 〈 a1Tk1 , a1Ik1 , a1Fk1 〉...〈 ank , ankI n , ankF n 〉 ) det ( B)) n

( k1 , k2 ...kn )∈Sn

= det ( A) det ( B ) (2) We know that det ( AB ) = det (( n

∈Sn

k =1

∑ (a k =1

...

(a ∨ b

∑ (a σ ì

k =1

t ≤ s ,t ≤ n

T 1σ (1)

k =1

(a ∨ b

T ns

∈Sn

k =1

T 1s

T tσ ( n )

t ≤ s ,t ≤ n

≤

k =1

∧ bkTσ ( n ) ), ∑ ( ankI ∧ bkIσ ( n ) ), ∏ ( ankF ∨ bkTσ ( n ) )]

∑(ì σ ì

k =1 n

k =1

∈Sn t ≤ s ,t ≤ n

(

∧ bkTσ (1) , ∑ a1Ik ∧ bkIσ (1) , ∏ a1Fk ∨ bkTσ (1) ...

T 1k

T nk

∑ aikT ∧ bkjT , ∑ aikI ∧ bkjI , ∏ aikF + bkjF ))

∑ [∑ a σ n

∨b

T tσ (1)

), (

T 1σ (1)

), (

(a ∨ btIσ (1) ),

t ≤ s ,t ≤ n

(a ∨ b

I ns

I tσ ( n )

t ≤ s ,t ≤ n

) ∧ (a

I 1σ (1)

I 1s

t ≤ s ,t ≤ n

(a1Fs ∧ bsFσ (1) )...

(ansF ∧ bsFσ ( n ) )

t ≤ s ,t ≤ n

∨b

I 1σ (1)

) ∧ (a1Fσ (1) ∧ b1Fσ (1) )

(anTσ ( n ) ∨ bnTσ ( n ) ) ∧ (anIσ ( n ) ∨ bnIσ ( n ) ) ∧ (anFσ ( n ) ∧ bnFσ ( n ) )

= det ((〈 aijT , aijI , aijF 〉 + 〈bijT , bijT , bijT 〉 ) n ) = det ( A + B ) r Corollary 3.10. Let A be a FNSSM, Ar = (aij ) ∈ FNSSM n (r=1, 2, 3,...,m}. Then

(1) det ( A1 ) det ( A2 )...det ( Am ) ≤ det (

∑ Ar ) where r =1

r =1

∑ Ar = (∑ aijr )n ∈ FNSSM n .

(2) det ( A( r ) ) = det ( A), where A = ( aij ) n ∈ FNSSM n and r ∈ N . Example 3.11. Consider the 4 × 4 matrix

 〈 a11T , a11I , a11F 〉 〈 a12T , a12I , a12F 〉  T I F T I F  〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉 T T  〈 a31 , a3I1 , a31F 〉 〈 a32 , a32I , a32F 〉  T I F T I F  〈 a41 , a41 , a41 〉 〈 a42 , a42 , a42 〉

〈 a14T , a14I , a14F 〉  T I F T I F  〈 a23 , a23 , a23 〉 〈 a24 , a24 , a24 〉 T I F T I F  〈 a33 , a33 , a33 〉 〈 a34 , a34 , a34 〉  T I F T I 〈 a43 , a43 , a43 〉 〈 a44 , a44 , a4F4 〉  〈 a13T , a13I , a13F 〉

We find the determinant of the above matrix in the following method

〈 a11T , a11I , a11F 〉

〈 a12T , a12I , a12F 〉

T I F T I F 〈 a21 , a21 , a21 〉 〈 a22 , a22 , a22 〉 1< 2

T T 〈 a33 , a33I , a33F 〉 〈 a34 , a34I , a34F 〉 T I F T I F 〈 a43 , a43 , a43 〉 〈 a44 , a44 , a44 〉

R.Uma, P. Murugadas and S. Sriram T T 〈 a32 , a32I , a32F 〉 〈 a34 , a34I , a34F 〉 〈 a11T , a11I , a11F 〉 〈 a13T , a13I , a13F 〉 + T I F T I F T I F T I F , a42 , a42 〉 〈 a44 , a44 , a44 〉 〈 a21 , a21 , a21 〉 〈 a23 , a23 , a23 〉 1<3 〈 a42

〈 a11T , a11I , a11F 〉

〈 a14T , a14I , a14F 〉

T 〈 a32 , a32I , a32F 〉

T 〈 a33 , a33I , a33F 〉

T I F T I F T I F T I F , a42 , a42 , a43 , a43 〉 〈 a43 〉 〈 a21 , a21 , a21 〉 〈 a24 , a24 , a24 〉 1<4 〈 a42

T T 〈 a31 , a31I , a31F 〉 〈 a34 , a34I , a34F 〉 〈 a12T , a12I , a12F 〉 〈 a13T , a13I , a13F 〉 + T I F T I F T I F T I F , a41 , a41 , a44 , a44 〉 〈 a44 〉 〈 a22 , a22 , a22 〉 〈 a23 , a23 , a23 〉 2<3 〈 a41

〈 a12T , a12I , a12F 〉

〈 a14T , a14I , a14F 〉

T I F T I F , a22 , a22 , a24 , a24 〈 a22 〉 〈 a24 〉 2< 4

T T 〈 a31 , a31I , a31F 〉 〈 a33 , a33I , a33F 〉 T I F T I F 〈 a41 〉 〈 a43 〉 , a41 , a41 , a43 , a43

T T 〈 a31 , a31I , a31F 〉 〈 a32 , a32I , a32F 〉 〈 a13T , a13I , a13F 〉 〈 a14T , a14I , a14F 〉 T I F T I F T I F T I F , a41 , a41 〉 〈 a42 , a42 , a42 〉 〈 a23 〉 〈 a24 〉 3< 4 〈 a41 , a23 , a23 , a24 , a24

using this method we can find the determinant of the given matrix

 〈 0.4, 0.2, 0.1〉 〈 0.6, 0.7, 0.8〉   〈 0.4,0.6, 0.7〉 〈 0.3, 0.2,0.1〉  〈 0.6, 0.7, 0.8〉 〈 0.8, 0.9, 0.3〉   〈 0.9,0.5, 0.3〉 〈 0.5, 0.3, 0.2〉

〈 0.7, 0.3, 0.4〉 〈 0.6, 0.7, 0.8〉   〈 0.5, 0.6, 0.7〉 〈 0.4, 0.3,0.2〉  〈 0.5, 0.6, 0.7〉 〈 0.6, 0.7, 0.8〉   〈 0.5, 0.6, 0.7〉 〈 0.8, 0.3, 0.9〉 

Solution. =

〈 0.4, 0.2, 0.1〉〈 0.6, 0.7,0.8〉 〈 0.5, 0.6, 0.7〉〈 0.6, 0.7, 0.8〉 〈 0.4, 0.6, 0.7〉〈 0.3, 0.2, 0.1〉 〈 0.5, 0.6, 0.7〉〈 0.8, 0.3, 0.9〉

〈 0.4, 0.2, 0.1〉〈 0.7, 0.3, 0.4〉 〈 0.8, 0.9, 0.3〉〈 0.6, 0.7, 0.8〉 〈 0.4, 0.6, 0.7〉〈 0.5, 0.6, 0.7〉 〈 0.5,0.3, 0.2〉〈 0.8, 0.3,0.9〉

〈 0.4, 0.2, 0.1〉〈 0.6, 0.7, 0.8〉 〈 0.8, 0.9, 0.3〉〈 0.5, 0.6, 0.7〉 〈 0.4, 0.6, 0.7〉〈 0.4, 0.3, 0.2〉 〈 0.5, 0.3, 0.2〉〈 0.5, 0.6, 0.7〉 +

〈 0.6, 0.7, 0.8〉〈 0.7, 0.3, 0.4〉 〈 0.6, 0.7, 0.8〉〈 0.6, 0.7, 0.8〉 〈 0.3, 0.2, 0.1〉〈 0.5, 0.6, 0.7〉 〈 0.9, 0.5, 0.3〉〈 0.8, 0.3, 0.9〉

〈 0.6, 0.7, 0.8〉〈 0.6, 0.7, 0.8〉 〈 0.6, 0.7, 0.8〉〈 0.5, 0.6, 0.7〉 〈 0.3, 0.2, 0.1〉〈 0.4, 0.3, 0.2〉 〈 0.9, 0.5, 0.3〉〈 0.5, 0.6, 0.7〉

〈 0.7, 0.3, 0.4〉〈 0.6, 0.7, 0.8〉 〈 0.6, 0.7, 0.8〉〈 0.8, 0.9, 0.3〉 〈 0.5, 0.6, 0.7〉〈 0.4, 0.3, 0.2〉 〈 0.9, 0.5, 0.3〉〈 0.5, 0.3,0.2〉 = [〈 0.3, 0.2, 0.1〉 ∨ 〈 0.4, 0.6, 0.8〉 ][〈 0.5, 0.3, 0.9〉 ∨ 〈 0.5, 0.6, 0.8〉 ] + [〈 0.4, 0.2, 0.7〉 ∨ 〈 0.4, 0.3, 0.7〉 ][〈 0.8, 0.3, 0.9〉 ∨ 〈 0.5, 0.3, 0.8〉 ] + [〈0.4, 0.2, 0.2〉 ∨ 〈 0.4, 0.6, 0.8〉 ][〈 0.5, 0.6, 0.7〉 ∨ 〈 0.5, 0.3, 0.7〉 ] + [〈 0.5, 0.6, 0.8〉 ∨ 〈 0.3, 0.2, 0.4〉 ][〈 0.6, 0.3, 0.9〉 ∨ 〈 0.6, 0.5.0.8〉 ] +

100

Determinant Theory for Fuzzy Neutrosophic Soft Matrices [ 〈 0.4, 0.3, 0.8〉 ∨ 〈 0.3, 0.2, 0.8〉 ][〈 0.5, 0.6, 0.8〉 ∨ 〈 0.5, 0.5, 0.7〉 ] + [ 〈 0.4, 0.3, 0.4〉 ∨ 〈 0.5, 0.6, 0.8〉 ][〈 0.5, 0.3, 0.8〉 ∨ 〈 0.8, 0.5, 0.3〉 ] = [〈 0.4, 0.6, 0.1〉 ∨ 〈 0.5, 0.6, 0.8〉 ][〈 0.4, 0.3, 0.7〉 ∨ 〈 0.8, 0.3, 0.8〉 ] + [〈0.4, 0.6, 0.2〉〈 0.5, 0.6, 0.7〉 ] + [〈 0.5, 0.6, 0.4〉〈 0.6, 0.5, 0.8〉 ] +

[〈0.4, 0.3, 0.8〉〈 0.5, 0.6, 0.7〉 ] + [〈 0.5, 0.6, 0.4〉〈 0.8, 0.5, 0.3〉 ] = 〈 0.4, 0.6, 0.8〉 + 〈 0.4, 0.3, 0.8〉 + 〈 0.4, 0.6, 0.7〉 + 〈 0.5, 0.5, 0.8〉 + 〈 0.4, 0.3, 0.8〉 + 〈 0.5, 0.5, 0.4〉 .

det ( A) = 〈 0.5, 0.6, 0.4〉

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