Determinant Theory for Fuzzy Neutrosophic Soft Matrices

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1 Progress i Noliear Dyamics ad Chaos Vol. 4, No. 2, 206, ISSN: (olie) Published o 30 November DOI: Progress i Determiat Theory for uzzy Neutrosophic Soft Matrices R.Uma, P. Murugadas 2 ad S. Sriram 3,2 Departmet of Mathematics, Aamalai Uiversity, Aamalaiagar , Idia 3 Departmet of Mathematics, Mathematics wig, D.D.E, Aamalai Uiversity, Aamalaiagar , Idia Emil: uma83bala@gmail.com; 3 ssm_3096@yahoo.co.i *Correspodig author. address: 2 bodi_muruga@yahoo.com Received 23 September 206; accepted 7 October 206 Abstract. We study determiat theory for fuzzy eutrosophic soft square matrices, its properties ad also we prove that det( Aadj( A)) = det( A) = det( adj( A) A). Keywords: Adjoit, determiat, fuzzy eutrosophic soft square matrix, determiat of fuzzy eutrosophic soft square matrix. AMS Mathematics Subject Classificatio (200): 0M20. Itroductio The complexity of problems i ecoomics, egieerig, evirometal scieces ad social scieces which caot be solved by the well kow methods of classical Mathematics pose a great difficulty i today s practical world (as various types of ucertaities are preseted i these problems). To hadle situatio like these, may tools have bee suggested. Some of them are probability theory, fuzzy set theory [8], rough set theory [], etc. The traditioal fuzzy set is characterized by the membership value or the grade of membership value. Sometimes it may be very difficult to assig the membership value for fuzzy sets. Iterval-valued fuzzy sets were proposed as a atural extesio of fuzzy sets ad the iterval valued fuzzy sets were proposed idepedetly by Zadeh [9] to ascertai the ucertaity of grade of membership value. I curret sceario of practical problems i expert systems, belief system, iformatio fusio ad so o, we must cosider the truth membership as well as the falsity-membership for proper descriptio of a object i imprecise ad doubtful eviromet. Neither the fuzzy sets or the iterval valued fuzzy sets is appropriate for such a situatio. Ituitioistic fuzzy set iitiated by Ataassov [3] is appropriate for such a situatio. The ituitioistic fuzzy sets ca oly hadle the icomplete iformatio cosiderig both the truth membership (or simply membership) ad falsity-membership (or o membership) values. It does ot hadle the idetermiate ad icosistet iformatio which exist i belief system. The soft set theory, a utterly ew theory for modelig ambiguity ad ucertaities was first coied by Molodstov [9] i the year

2 R.Uma, P. Murugadas ad S. Sriram Soft set theory research is carried out as a ew tred ad it shows much appreciable developmet well received by the users of the field. uzzy matrices play crucial role i Sciece ad Techology. Sometimes the issues caot be solved by classical matrix theory whe they occur i a ucertai eviromet ad this failure is ievitable. Thomaso [4] iitiated the fuzzy matrices to represet fuzzy relatio i a system based o fuzzy set theory ad discussed about the covergece of power of fuzzy matrix. I 995, Smaradache itroduced the cocept of eutrosophy. I eutrosophic logic, each propositio is approximated to have the percetage of truth i a subset T, the percetage of idetermiacy i a subset I ad the percetage of falsity i a subset, so that this eutrosophic logic is called a extesio of fuzzy logic. I fact this mathematical tool is used to hadle problems like imprecisio, idetermiacy ad icosistecy of data etc. Maji et al. [5], iitiated the cocept of fuzzy soft set with some properties regardig fuzzy soft uio, itersectio, complemet of fuzzy soft set. Moreover Maji et al. [6,0] exteded soft sets to ituitioistic fuzzy soft sets ad eutrosophic soft sets ad the cocept of eutrosophic set was itroduced by Smaradache [2] which is a geeralizatio of fuzzy logic ad several related systems. Yag ad Ji [7], itroduced a matrix represetatio of fuzzy soft set ad applied it i decisio makig problems. Bora et al. [8] itroduced the ituitioistic fuzzy soft matrices ad applied i the applicatio of a Medical diagosis. Sumathi ad Arokiarai [3] itroduced ew operatio o fuzzy eutrosophic soft matrices. Dhar et al. [7] have also defied eutrosophic fuzzy matrices ad studied square eutrosophic fuzzy matrices. Uma et al. [5,6], itroduced two types of fuzzy eutrosophic soft matrices ad have discussed determiat ad adjoit of fuzzy eutrosophic soft matrices. Kim et al. [4], itroduced the cocept of determiat theory for fuzzy matrices. I this paper, some elemetary properties of determiat theory for fuzzy eutrosophic soft square matrices have bee established ad some theorems icludig det( A( adja)) = det( A) = det( adj( A) A). where det( A ) deotes the determiat of A ad adj( A ) deotes the adjoit matrix of A. 2. Prelimiaries Defiitio 2.. [2] A eutrosophic set A o the uiverse of discourse X is defied as A = x, T ( x), I ( x), ( x), x X, { } A A A where T, I, : X ] 0, [ ad 0 TA( x) I A( x) A ( x) 3 () rom philosophical poit of view the eutrosophic set takes the value from real stadard or o-stadard subsets of ] 0, [. But i real life applicatio especially i scietific ad Egieerig problems it is difficult to use eutrosophic set with value from real stadard or o-stadard subset of ] 0, [. Hece we cosider the eutrosophic set which takes the value from the subset of [0,].Therefore we ca rewrite the equatio () as 0 T ( x) I ( x) ( x) 3. A A A 86

3 Determiat Theory for uzzy Neutrosophic Soft Matrices I short a elemet aɶ i the eutrosophic set A, ca be writte as T I T I aɶ = a, a, a, where a deotes degree of truth, a deotes degree of idetermiacy, a deotes degree of falsity such that 0 a T a I a 3. Example 2.2. Assume that the uiverse of discourse X = { x, x2, x3}, where x, x 2, ad x 3 characterizes the quality, reliability, ad the price of the objects. It may be further assumed that the values of { x, x2, x 3} are i [0,] ad they are obtaied from some ivestigatios of some experts. The experts may impose their opiio i three compoets viz; the degree of goodess, the degree of idetermiacy ad the degree of pooress to explai the characteristics of the objects. Suppose A is a Neutrosophic Set (NS) of X, such that A = { x,0.4,0.5,0.3, x2,0.7,0.2,0.4, x3,0.8,0.3,0.4 }, where for x the degree of goodess of quality is 0.4, degree of idetermiacy of quality is 0.5 ad degree of falsity of quality is 0.3, etc. Defiitio 2.3. [9] Let U be a iitial uiverse set ad E be a set of parameters. Let P(U) deotes the power set of U. Cosider a oempty set A, A E. A pair (,A) is called a soft set over U, where is a mappig give by : A P( U ). Defiitio 2.4. [] Let U be a iitial uiverse set ad E be a set of parameters. Cosider a o empty set A, A E. Let P( U ) deotes the set of all fuzzy eutrosophic sets of U. The collectio (, A ) is termed to be the uzzy Neutrosophic Soft Set (NSS) over U, Where is a mappig give by : A P( U ). Hereafter we simply cosider A as NSS over U istead of (, A ). Defiitio 2.5. [2] Let U = { c, c2,... c m } be the uiversal set ad E be the set of { } parameters give by E = e, e2,... e. Let A E. A pair (, A ) be a NSS over U. The the subset of U E is defied by R = {( u, e); e A, u ( e)} which is called A a relatio form of ( A, E ). The membership fuctio, idetermiacy membership fuctio ad o membership fuctio are writte by T : U E [0,], I : U E [0,] ad : U E [0,] where T ( u, e) [0,], I ( u, e) [0,] R A R A ad R A ( u, e) [0,] are the membership value, idetermiacy value ad o membership value respectively of u U for each e E. If [( T, I, )] = [( T ( u, e ), I ( u, e ), ( u, e )] we defie a matrix ij ij ij ij i j ij i j ij i j RA A R A RA 87

4 R.Uma, P. Murugadas ad S. Sriram T, I, T2, I2, 2 T, I, T2, I2, 2 T22, I22, 22 T2, I2, 2 Tij, Iij, ij = m Tm, Im, m Tm 2, Im2, m 2 Tm, Im, m This is called a m NSM of the NSS ( A, E) over U. Defiitio 2.6. [5] Let U = { c, c2... c m } be the uiversal set ad E be the set of parameters give by E = { e, e2,... e }. Let A E. A pair (, A ) be a fuzzy eutrosophic soft set. The fuzzy eutrosophic soft set (, A ) i a matrix form as A = ( a ) or A = ( a ), i =, 2,... m, j =, 2,... where m ij m ij ( T ( ci, e j ), I ( ci, e j ), ( ci, e j )) if e j A ( aij ) = 0,0, if e j A where Tj ( c i ) represet the membership of ci, I j ( c i ) represet the idetermiacy of c i ad j ( c i ) represet the o-membership of c i i the NSS (, A ). If we replace the idetity elemet 0, 0, by 0,, i the above form we get NSM of type-ii. Let m deotes NSM of order m ad deotes NSM of order. Defiitio 2.7. [5] [Type-I] Let A = ( aij, aij, aij ), B = ( bij, bij, b ij ) m the compoet wise additio ad compoet wise multiplicatio is defied as T T I I A B = ( sup a, b, sup a, b, if a, b ).. { ij ij } { ij ij} { ij ij } 2. A B = ( if { a T, b T }, if { a I, b I }, sup{ a, b }). ij ij ij ij ij ij Defiitio 2.8. [5] Let A, B, the compositio of A ad B is defied as m p T T I I A B = ( aik bkj ), ( aik bkj ), ( aik bkj ) k = k= k= equivaletly we ca write the same as ( ( ), ( ), T T I I ( a b a b a b ) k = ik kj k = ik kj k = ik kj ) = The product A B is defied if ad oly if the umber of colums of A is same as the umber of rows of B. A ad B are said to be coformable for multiplicatio. We shall use AB istead of A B. Defiitio 2.9.[5][Type-II] Let A = ( aij, aij, aij ), B = ( bij, bij, b ij ) m, the compoet wise additio ad compoet wise multiplicatio is defied as 88

5 Determiat Theory for uzzy Neutrosophic Soft Matrices T T I I { ij ij } { ij ij} { ij ij } A B = ( sup a, b, if a, b, if a, b ). T T I I A B = ( if { a, b }, sup{ a, b }, sup{ a, b } ). ij ij ij ij ij ij Aalogous to NSM of type-i, we ca defie NSM of type -II i the followig way T I Defiitio 2.0. [5] Let A = ( aij, aij, aij ) = ( a ij ) m ad B = ( b T, b I, b ) = ( b ) the product of A ad B is defied as ij ij ij ij p T T I I A B = aik bkj, aik bkj, aik bkj k = k = k= equivaletly we ca write the same as T T I I = aik bkj, aik bkj, aik bkj. k = k = k = The product A B is defied if ad oly if the umber of colums of A is same as the umber of rows of B. A ad B are said to be coformable for multiplicatio. Defiitio 2.. [6] The determiat A of NSM A = ( a T, a I. a ) is defied 89 ij ij ij as follows T A = a T... a I, a I... a, a... a σ () σ ( ) σ () σ ( ) σ () σ ( ) σ S σ S σ S where S deotes the symmetric group of all permutatios of the idices (, 2,... ). Defiitio 2.2. [6] The adjoit of a NSM A deoted by adj A, is defied as follows bij = Aji is the determiat of the ( ) ( ) NSM formed by deletig row j ad colum i from A ad B = adja. Remark 2.3. We ca write the elemet b ij of adja = B = ( b ij ) as follows: b a, a, a = Where { } T I ij tπ ( t) tπ ( t) tπ ( t) π S j t i j all permutatio of set j over the set i. =, 2,3... \{ j} ad j S j is the set of i 3. Properties of the fuzzy eutrosophic soft square matrices (NSSM). The value of the determiat remais uchaged whe ay two rows or colums are iterchaged. 2. The values of the determiat of NSSM remai uchaged whe rows ad colums are iterchaged. 3. If A ad B be two NSSMs the the followig property will hold det( AB) deta detb. 4. If the elemets of ay row (or colum) of a determiat are added to the correspodig elemets of aother row (or colum), the value of the determiat thus obtaied is equal to the value of the origial determiat.

6 R.Uma, P. Murugadas ad S. Sriram Theorem 3.. Let A = ( a T, a I, a ) NSSM. ij ij ij A = ( a, a, a, a, a, a,..., a, a, a ) be the k -th row of A. We T I Let k k k k k 2 k 2 k 2 k k k assume that a = a T, I, ki aki aki for all i, 2,..., ad a pq a for all p, q, 2,...,. The det( A) = a. Theorem 3.2. Let A NSSM the T I (i) det( A) = A = a, a, a A, i {, 2,..., }. t= it it it it a e, a e, a e a f, a f, a 2 (ii) det( A) = A, where the summatio is e< f a2e, a2e, a2e a2 f, a2 f, a2 f e f take over all e ad f i {,2,...,} such that e < f. Defiitio 3.3. Let A = ( a T, a I, a ) NSSM, ad let B be a matrix from A by ij ij ij strikig out e, row e 2,..., row e k ad colum g, colum g, 2..., colum g k. we defie e e2... ek A = det( H ). g g2... gk Theorem 3.4. T I T T T a g, a g, a g... a gk, a gk, a gk a2 g, a2 g2, a2 g2... a2 gk, a2 gk, a2 gk det( A) = det g < g2 <... < g k T I k T I ak g, ak g, a g... ak gk, ak gk, ak g k 2... k A where the summatio is take over all g g g 2... g, g2,..., gk {, 2,... }, k such that g < g2 <... < g k. Proof: Let S( g, g2,..., gk ) = { σ :{,2,..., k} { g, g2,..., gk} σ is a bijectio}. The det( A) = a, a, a... a, a, a g < g2 <... < gk σ S σ () σ () σ () σ ( ) σ ( ) σ ( ) = ( ) a, a, a... a, a, a ) {,2,..., k} S ( g, g2,..., gk () () () ( ) ( ) ( ) σ σ σ σ σ σ σ 90

7 g < g2 <... < gk Determiat Theory for uzzy Neutrosophic Soft Matrices = ( )( a, a, a... a, a, a ) 2... k A g g... g k ' {,2,..., k} S ( g, g2,..., gk ) ' () ' () ' () ' ( ) ' ( ), ( ) σ σ σ σ σ σ σ = 2 g < g2 <... < gk T I a g, a g, a g a g2, a g2, a g2... a gk, a gk, a gk T I a2 g, a2 g, a2 g a2 g2, a2 g2, a2 g2... a2 gk, a2 gk, a2 gk T I ak g, ak g, ak g ak g2, ak g2, ak g2... ak gk, ak gk, ak gk Hece the pooof k A g g 2... g k Lemma 3.5. a, a, a b, b, b Let A = be a NSSM. c, c, c d, d, d a, a, a b, b, b c, c, c d, d, d The det det a, a, a b, b, b c, c, c d, d, d a, a, a b, b, b c, c, c d, d, d = det( A) a, a, a b, b, b c, c, c d, d, d Proof: a, a, a b, b, b c, c, c d, d, d We see that det det a, a, a b, b, b c, c, c d, d, d = a, a, a b, b, b c, c, c d, d, d a, a, a d, d, d b, b, b c, c, c = det( A). Notatio: Let A NSSM. Let A( e f ) be the matrix obtaied from A by replacig row f of A by row e of A. Theorem 3.6. Let A NSSM. The ( i) det( A(2 )) det( A( 2)) det( A). ( ii) det( A(2 )) det( A(3 2)) det( A). ( iii) det( A( q p))) det( A( p k)) det( A). Proof: To prove det 9

8 2 2 2,, 2, 2, 2 R.Uma, P. Murugadas ad S. Sriram ( i) det( A(2 )) det( A( 2)) = A a a a a a a 2 A... e f a, a, a a, a, a a, a, a a, a, a A a2e, a2e, a2e a2 f, a2 f, a2 A 2... a e, a e, a e a f, a f, a e f , 2, 2 2, 2, 2 e< f a a a a a a 2 A 92 e e e f f f e, e, e f, f, f a a a a a a e< f a a a a a a A , 2, 2 22, 22, 22 a e, a e, a e a f, a f, a = A 2 a2 g, a2 g, a2 g a2h, a2h, a2h A e< f a e, a e, a e a f, a f, a e f g< h a2 g, a2 g, a2 g a2h, a2h, a2h 2 g h a e, a e, a e a f, a f, a A 2 e< f a2 g, a2 g, a2 g a2h, a2h, a2h e f A 2 (by Lemma 3.5) g h < g h e f We ow itroduce symbols J, J 2, A ad J. Defie g h e f a e, a e, a e a f, a f, a J = g h a2g, a2g, a2g a2h, a2h, a2h A 2 e f A 2 g h e f e f J = J = J, ( e, f ) = ( g, h) g h e< f e f e f J 2 = J ad J = J J. 2 The we see that ( e, f ) ( g, h) g h 2 a, a, a a2, a2, a2 J = a2, a2, a2 a22, a22, a22 A 2, J = det( A) by Theorem 3.2(ii) ad det( A(2 )) det( A( 2)) J = J J = det( A) J. 2 2

9 Determiat Theory for uzzy Neutrosophic Soft Matrices We show that J 2 det( A). There are two cases to be cosidered. 2 Case. We cosider a = J, a term of 3 J. 2 Let a = a T, a I, a T a23, a I 23, a 23 A 2 A 2 ad 3 2 a2 = ( a T 2, a I 2, a T 2 a2, a I 2, a 2 A A 2 3 The a = a a, 2 a a, a, a a3, a3, a3 a2, a2, a2 a23, a23, a23 A 3 3 det( A), a, a, a a2, a2, a2 2 a2 A det ( A ), a2, a2, a2 a22, a22, a22 2 ad J det ( A ), 3 2 Case 2. We take J 2 2 Let b = a, a, a a2, a2, a2 A A ad 2 2 b2 = a2, a2, a2 a2, a2, a2 A A. 2 The J = b b 2. To show that b det( A) ad b2 = det( A), we 2 observe all coordiates of the elemets a ij ivolved i A 2 ad A. The coordiates of the elemets a ij ivolved i these determiats are all coordiates of the elemets of the k th row A k of A, for k 3. Therefore, if we let b = a a... a... a, the we see that k 2 k 2 b ( a, a, a a2, a2, a2 ) c det( A). or b, let 2 c = a3a4 2a a 3a2, the we see that b2 ( a2, a2, a2 a2, a2, a2 ) c det( A). or ay e f J,we apply either the case or the case 2 ad we ca deduce that g h ( e, f ) ( g, h) 93

10 R.Uma, P. Murugadas ad S. Sriram e f J det( A). g h ( e, f ) ( g, h) Thus (i) holds. (ii). irst we cosider b, b, b b, b, b b, b, b b, b, b , 3, 3 32, 32, 32 a a a a a a , 2, 2 22, 22, 22 b b b b b b b2, b2, b2 b32, b32, b32. a3, a3, a3 a32, a32, a32 We itroduce a symbol K g h a2 g, a2g, a2g a2h, a2h, a2h = e f a3e, a3e, a3e a3 f, a3 f, a3 The we ca see that det( A(2 )) det( A(3 2)) a e, a e, a e a f, a f, a = A 2 3 e< f a3e, a3e, a3e a3 f, a3 f, a3 e f < g h A A. g h e f 2 3 g h a2g, a2 g, a2 g a2h, a2h, a2h A A g< h a3e, a3e, a3e a3 f, a3 f, a3 g h e f = e< f g< h e< f g A e h f a a a a a a A 2g 2g 2g 2h 2h 2h 2g, 2g, 2g 2h, 2h, 2h = g h K g h ( g, h) = ( e, f ) e f K. ( g, h) ( e, f ) e f g h Next we prove that K det( A). e f ( g, h) ( e, f ) or this we cosider two cases. 2 Case. We take K. We see that 3 K 2 =( a T 2, a I 2, a 2 a T 33, a I 33, a 33 a T 22, a I 22, a T 22 a3, a I 3, a 3 ) A 2 3 A

11 Determiat Theory for uzzy Neutrosophic Soft Matrices = ( a2, a2, a2 a33, a33, a33 ) A 2 3 A 2 3 ( a T 22, a I 22, a 22 a T 3, a I 3, a 3 ) A A a2, a2, a2 a23, a23, a23 A 2 3 a2, a2, a2 a22, a22, a22 a3, a3, a3 a33, a33, a33 3 a3, a3, a3 a32, a32, a32 A 2 3 det( A) det( A) = det( A ). Case 2. We cosider K = a 2, a 2, a 2 a 32, a 32, a 32 A A 2 3 a2, a2, a2 a3, a3, a3 A A Cosiderig the coordiates of the elemets a ij ivolved i A 2 A 2 3, we claim that ( a T 2, I 2, T 2 32, I 32, a a a a a32 ) A A 2 3 det( A) ad ( a T 2, I 2, T 2 3, I 3, a a a a a3 A A 2 3 det( A). Similarly we ca prove (iii). Theorem 3.7. Let A = ( a T, a I, a ), B = ( b T, b I, ),( c T, c I, c ) NSSM. The ij ij ij ij ij ij ij ij ij. If a, a, a a, a, a ( k =, 2,..., ) for all i, the ii ii ii ik ik ik det( A) = a, a, a a, a, a... a, a, a. T T T T T T T T T A C 2. det det( A) det( B ) where O = ( 0,0, ) NSSM O B T 3. det( AA ) det( A). 4. If a T, a I, a b T, b I, b for all i, j, the det( A) = det( B). ij ij ij ij ij ij Proof:. We have T I T I a, a, a a, a T I, a... a, a T, a a I, a, a for every σ, σ () 2 σ (2) σ ( ) sice a T, a I, a a T, a I, a ( k =, 2,..., ) for all i. ii ii ii ik ik ik S 95

12 R.Uma, P. Murugadas ad S. Sriram det( A) = a, a, a a, a, a... a, a, a T I Hece σ () σ () σ () 2 σ (2) 2 σ (2) 2 σ (2) σ ( ) σ ( ) σ ( ) σ S a, a, a a, a, a... a, a, a T I = A C 2. Let = ( d T, I, ij dij dij ) 2. O B The A C det = d σ (), d σ (), d σ ()... d2 σ (2 ), d2 σ (2 ), d2 σ (2 ) O B σ S2 = d σ (), d σ (), d σ ()... d2 σ (2 ), d2 σ (2 ), d2 σ (2 ) σ = σ σ S2, ( i) (if i ) σ S2, k>, if ( k ) σ d, d, d... d, d, d σ () σ () σ () 2 σ (2 ) 2 σ (2 ) 2 σ (2 ) d, d, d... d, d, d 0 σ () σ () σ () 2 σ (2 ) 2 σ (2 ) 2 σ (2 ) S2, σ ( i) ( ifi ) T I T I d σ '(), d σ '(), d σ '()... d σ '( ), d σ '( ), d σ '( ) det( B) ' S = σ ( d, d, d... d, d, d ) det( B) = σ () σ () σ () σ ( ) σ ( ) σ ( ) σ S = det( A) det( B ). 3. Let T ( T, I, T I AA = g g g ), where g, g, g = a, a, a a, a, a. ij ij ij We have, for every σ S T I g, g, g g, g, g... g, g, g = ( a k, a k, a k )...( ak, ak, ak ) k = k = a σ (), a σ (), a σ ()... a σ ( ), a σ ( ), a σ ( ) ij ij ij ik ik ik kj kj kj k = det( AA ) g, g, g g, f, f... g, g, g T T I Hece a, a, a... a, a, a σ S = det( A). σ () σ () σ () σ ( ) σ ( ) σ ( ) Theorem 3.8. Let A = ( a ij ) be a NSSM. The we have the followig det( Aadj( A)) = det( A) = det( adj( A) A). Proof: We prove that det( Aadj( A)) = det( A). We first cosider =2. 96

13 Determiat Theory for uzzy Neutrosophic Soft Matrices a, a, a a2, a2, a2 Let A =. a2, a2, a2 a22, a22, a22 The we see that a22, a22, a22 a2, a2, a2 adj( A) =. a2, a2, a2 a, a, a det( A) a, a, a a2, a2, a2 det( Aadj( A)) = a, a, a a, a, a det( A) det( A) ( a, a, a a, a, a a, a, a a, a, a ) det( A). Next cosider > 2. We ca see that Aadj( A ) = T I a t, a t, a t A t a t, a t, a t A2 t... a t, a t, a t A t T I a2t, a2t, a2t A t a2t, a2t, a2t A2 t... a2t, a2t, a2t At T I at, at, a3t A t at, at, at A2 t... at, at, at A t = T I a, a, a A, T I T I = ( it it it jt ) det( Aadj ( A)) = ( a, a, a A )( a, a, a A )...( a, a, a A ). π S T I t t t π () t 2t 2t 2 t π (2) t t t t π ( ) t Clearly ay diagoal etry of the matrix Aadj( A ) is equal to det( A ). We prove the result i the followig way. () Let us defie T I T = ( a, a, a A )( a, a, a A )...( a, a, a A ), π t t t π () t 2t 2t 2 t π (2) t t t t π ( ) t π S Let e be the idetity of the group S. If e, T for. π = the = π det( A). Suppose that there exists k {, 2,..., } such that π ( k) = k. The we see that a a a A = a a a A kt kt kt π ( k ) t kt kt kt kt = det( A ) ad J = ( a, a, a A )( a, a, a A )... det( A)...( a, a, a A ) T I π t t t π () t 2t 2t 2 t π (2) t t t t π ( ) t det( A). (2) Let π be a permutatio i S. Assume that π ( k) k for all k {, 2,... }. We kow that every permutatio π ca be writte as a product of disjoit cycles π i ad let π = π.... π 2 π k We further assume that π = ( 2), a traspositio. T I The J π has two factors, ( a, a t t, a t A () t ) ad π T I ( a 2, a t 2t, a 2 t A (2) t ), ad from these we see that π 97

14 R.Uma, P. Murugadas ad S. Sriram ( a, a, a A )( a, a, a A ) = ( a, a, a A )( a, a, a A ) t t t π () t 2t 2t 2 t π (2) t t t t 2t 2t 2t 2t t = det( A(2 )) det( A( 2)) det( A) (by theorem3. 2(i)) (3) If π = π... π 2 π k ad π ( s, t), the we ca prove that J det π ( A ) by a argumet used i(2). Cosider J π for π = π.... π 2 π k If π = ( k, e, f,...), the we see that J = ( a, a, a A )( a, a, a A )... π kt kt kt π ( k ) t) et et et π ( e) t) = ( a, a, a A )( a, a, a A )... = det( A( e k)) det( A( f e))... kt kt kt et et et et ft rom Theorem 3.6(iii), we obtai that det( A( e k)) det( A( f e)) det( A) ad cosequetly that J det π ( A ). This proves that det( Aadj( A)) = det ( A ). Similarly, we ca prove that det( adj( A) A) = det( A). Hece the proof. Theorem 3.9. Let A, B NSSM. The () det( AB) det( A) det( B). (2) det( AB) det( A B), where A B = ( sup{ a T, b T }, sup{ a I, b I }, if { a, b }) Proof. ij ij ij ij ij ij T T I I det AB = det aik bkj aik bkj aik bkj k = k = k = T T I I = a k bk σ () a k bk σ () aik bk σ () σ S k = k = k = ( ) ((,, )) {[ ],, ),, [ ]} T T I I ak bkσ ( ), ak bkσ ( ), ( ak bkσ ( ) k = k = k = = ( ( a a... a b b... b ) σ S k, k2... k k, k2... k T T T T T T k 2 k2 k kσ () k2σ (2) kσ ( ) ( ( a a... a b b... b ) I I I I I I k 2 k2 k kσ () k2 σ ( 2) kσ ( ) ( a a... a b b... b ) k 2k2 k k σ () k2 σ ( 2) kσ ( ) k, k2... k a k, a k, a k... a,, k a k a k { k, k2... k} S 98

15 Determiat Theory for uzzy Neutrosophic Soft Matrices bk σ () bk σ () bk σ () bk σ ( ) bk σ ( ) bk σ ( ),,...,, ) = σ S ( T I a, a T, a... a I, a, a ) det( B)) ( k, k2... k ) S = det( A) det( B) k k k k k k T T I I (2) We kow that det( AB) = det(( a b, a b, a b )) ik kj ik kj ik kj k = k = k = T T I I T = [ a k bk σ (), a k bk σ (), a k bk σ ()... σ S k = k = k= T T I I T ak bkσ ( ) ak bkσ ( ) ak bkσ ( ) k = k = k = ( ), ( ), ( )] ì ì = ( ( a b ),( ( a b ), ( a b )... T T I I s tσ () s tσ () s sσ () σ S t s, t t s, t t s, t ì ì ( ( a b ),( ( a b ), ( a b ) T T I I s tσ ( ) s tσ ( ) s sσ ( ) t s, t t s, t t s, t ( a b ) ( a b ) ( a b ) σ S ì T T I I σ () σ () σ () σ () σ () σ () ( a b ) ( a b ) ( a b )... T T I I σ ( ) σ ( ) σ ( ) σ ( ) σ ( ) σ ( ) t s, t = det(( a T, a I, a b T, b T, b T ) ) = det( A B) ij ij ij ij ij ij ê r Corollary 3.0. Let A be a NSSM, A = ( a ) NSSM (r=, 2, 3,...,m}. The ê r ij m m m () det( A ) det( A2 )... det( Am ) det( A r ) where r Ar = ( aij ) NSSM. r= r= r= ( r) (2) det( A ) = det( A), where A = ( a ) NSSM ad r N. ij Example 3.. Cosider the 4 4 matrix a, a, a a2, a2, a2 a3, a3, a3 a4, a4, a4 a2, a2, a2 a22, a22, a22 a23, a23, a23 a24, a24, a24 a3, a3, a3 a32, a32, a32 a3 3, a33, a33 a34, a34, a34 a4, a4, a4 a42, a42, a42 a43, a43, a43 a44, a44, a 44 We fid the determiat of the above matrix i the followig method a, a, a a2, a2, a2 a33, a33, a33 a34, a34, a34 = a2, a2, a2 a22, a22, a 22 a < 2 43, a43, a43 a44, a44, a44 99

16 a, a, a a3, a3, a3 < a, a, a a4, a4, a4 < a2, a2, a2 a3, a3, a3 < a2, a2, a2 a4, a4, a4 < R.Uma, P. Murugadas ad S. Sriram , 42, 42 44, 44, 44 a a a a a a , 42, 42 43, 43, 43 a a a a a a , 4, 4 44, 44, 44 a a a a a a , 4, 4 43, 43, 43 a a a a a a a3, a3, a3 a4, a4, a4 a3, a3, a3 a32, a32, a32 a23, a23, a23 a24, a24, a 24 a 3< 4 4, a4, a4 a42, a42, a42 usig this method we ca fid the determiat of the give matrix 0.4, 0.2, , 0.7, , 0.3, , 0.7, ,0.6, , 0.2,0. 0.5, 0.6, , 0.3, , 0.7, , 0.9, , 0.6, , 0.7, ,0.5, , 0.3, , 0.6, , 0.3,0.9 Solutio. 0.4,0.2,0. 0.6,0.7, , 0.6, , 0.7, 0.8 = 0.4, 0.6, , 0.2, , 0.6, , 0.3, , 0.2, , 0.3, , 0.6, , 0.6, , 0.2, , 0.7, , 0.6, , 0.3, , 0.9, , 0.7, ,0.3, , 0.3, , 0.9, , 0.6, , 0.3, , 0.6, ,0.7, ,0.3, , 0.7, , 0.7, , 0.2, , 0.6, , 0.5, , 0.3, , 0.7, , 0.7, , 0.2, , 0.3, , 0.7, , 0.6, , 0.5, , 0.6, , 0.3, , 0.7, , 0.7, , 0.9, , 0.6, , 0.3, , 0.5, , 0.3,0.2 = [ 0.3,0.2,0. 0.4,0.6,0.8 ][ 0.5, 0.3, ,0.6,0.8 ] [ 0.4,0.2, ,0.3,0.7 ][ 0.8,0.3, ,0.3,0.8 ] [ 0.4,0.2, ,0.6,0.8 ][ 0.5,0.6, ,0.3,0.7 ] [ 0.5,0.6, ,0.2, 0.4 ][ 0.6,0.3, , ] 00

17 Determiat Theory for uzzy Neutrosophic Soft Matrices [ 0.4,0.3, ,0.2,0.8 ][ 0.5,0.6, ,0.5,0.7 ] [ 0.4,0.3, ,0.6,0.8 ][ 0.5,0.3, ,0.5,0.3 ] =[ 0.4, 0.6, ,0.6,0.8 ][ 0.4, 0.3, ,0.3,0.8 ] [ 0.4,0.6, ,0.6,0.7 ] [ 0.5,0.6, ,0.5,0.8 ] [ 0.4,0.3, ,0.6,0.7 ] [ 0.5,0.6, ,0.5,0.3 ] = 0.4, 0.6, , 0.3, , 0.6, , 0.5, , 0.3, , 0.5, 0.4. det( A ) = 0.5, 0.6, Coclusio I this paper, we have studied properties of determiat ad adjoit of fuzzy eutrosophic soft square matrices. REERENCES. I.Arockiarai, I.R.Sumathi ad M.Jecy, uzzy Neutrosophic soft topological spaces, IJMA, 0 (203) I.Arockiarai ad I.R.Sumathi, A fuzzy eutrosophic soft Matrix approach i decisio makig, JGRMA, 2 (2) (204) K.Ataassov, Ituitioistic fuzzy sets, uzzy Sets ad Systems, 20 (986) J.B.Kim, A.Baartmas ad N.Sahadi, Determiat theory for fuzzy matrix, uzzy Sets ad Systems, 29 (989) P.K.Maji, R.Biswas ad A.R.Roy, uzzy Soft set, The Joural of uzzy Mathematics, 9 (3) (200) P.K.Maji, R.Biswas ad A.R.Roy, Ituitioistic fuzzy soft sets, Joural of uzzy Mathematics, 2 (2004) M.Dhar, S.Broumi ad.smaradache, A ote o square eutrosophic fuzzy matrices, Neutrosophic Sets ad Systems, 3 (204) M.Bora, B.Bora ad T.J.Neog ad D.K.Sut, Ituitioistic fuzzy soft matrix theory ad its applicatio i medical diagosis, Aals of uzzy Mathematics ad Iformatics, 7() (203) D.Molodtsov, Soft set theory first results, Computer ad mathematics with applicatios, 37 (999) P.K.Maji, Neutrosophic soft set, Aals of fuzzy mathematics ad iformatics, 5 (203) Z.Pawlak, Rough sets, Iteratioal joural of computig ad iformatio scieces, (982) Smaradache, Neutrosophic set, a geeralisatio of the ituitioistic fuzzy set, Iter. J. Pure Appl. Math., 24 (2005) I.R.Sumathi ad I.Arockiarai, New operatios o fuzzy eutrosophic soft matrices, Iteratioal Joural of Iovative Research ad Studies, 3 (3) (204) M.G.Thomas, Covergece of powers of a fuzzy matrix, Joural. Math. Aal. Appl., 57 (977) R.Uma, P.Murugadas ad S.Sriram, uzzy eutrosophic soft matrices of type-i ad type-ii, Commuicated. 0

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