# On Refined Neutrosophic Matrices and Their Application in Refined Neutrosophic Algebraic Equations

Authors: Mohammad Abobala
Journal: Journal of Mathematics
 (2021)

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## Abstract

The objective of this paper is to introduce the concept of refined neutrosophic matrices as matrices such as multiplication, addition, and ring property. Also, it determines the necessary and sufficient condition for the invertibility of these matrices with respect to multiplication. On the contrary, nilpotency and idempotency properties will be discussed.


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## Body

## 1. Introduction

Neutrosophy is a new branch of generalized logic found by Smarandache to deal with indeterminacy in all fields of human knowledge. Neutrosophic sets were applicable in decision-making [1], number theory [2, 3], and space theory [4, 5].The concept of refined neutrosophic structure was supposed firstly in [6] by splitting indeterminacy I into two levels of subindeterminacies I1andI2. This idea was used in the study of refined neutrosophic rings [7–9], modules [10, 11], and groups [6]. Recently, the concept of n-refined neutrosophic structures was defined and used in [12–14].Neutrosophic matrices were a useful tool to deal with indeterminacy in many studies; we find their basic definition and properties such as ring structure, multiplication, and other applications in [15, 16].Through this work, we define, for the first time, the concept of refined neutrosophic matrices as a direct application of the refined neutrosophic set. Also, we determine the necessary and sufficient condition for the invertibility of these matrices with many related examples. On the contrary, we build an example to show how refined matrices can be used in refined neutrosophic equations defined in [17].All refined neutrosophic matrices through this paper are defined over a neutrosophic fieldFI1,I2.The structure of refined neutrosophic numbers is taken asa+bI1+cI2 instead of a,bI1,cI2. This representation is based on the theory of n-refined neutrosophic rings proposed in [12], where refined neutrosophic numbers can be represented by this form without any loss of generality or algebraic properties.


## 2. Preliminaries

Definition 1.
(see [7]). Let K be a field, the neutrosophic field generated by K∪I, which is denoted by KI=K∪I.Definition 2.
(see [7]). Classical neutrosophic number has the form a+bI, where aandb are real or complex numbers and I is the indeterminacy such that 0⋅I=0 and I2=I which results in In=I for all positive integers n.Definition 3.
(neutrosophic matrix; see [15]). Let Mm×n=aij:aij∈KI, where KI is a neutrosophic field. We refer this to be the neutrosophic matrix.Remark 1.
(see [6]). The element I can be split into two indeterminacies I1andI2 with conditions(1)I12=I1,I22=I2,I1I2=I2I1=I1.Definition 4.
(see [1]). If X is a set, then XI1,I2=a,bI1,cI2:a,b,c∈X is called the refined neutrosophic set generated by X, I1,andI2.Definition 5.
(see [7]). Let (R, + , ×) be a ring; RI1,I2,+,× is called a refined neutrosophic ring generated by R, I1,andI2.Theorem 1 (see [7]).
LetRI1,I2,+,× be a refined neutrosophic ring; then, it is a ring.
It is called a neutrosophic field ifR is a classical field.Theorem 2.
(see [17]). Let A1X1+⋯+AnXn=C,C=c0,c1I1,c2I2,Xi=x0i,x1iI1,x2iI2,andAi=a0i,a1iI1,a2iI2, be a linear equation with n-variables over a refined neutrosophic field FI1,I2. Then, it is equivalent to the following system of classical linear equations over the classical field F:(a)
∑i=1na0ix0i=c0(b)
∑i=1na0i+a2ix0i+x2i=c0+c2(c)
∑i=1na0i+a1i+a2ix0i+x1i+x2i=c0+c1+c2


## 3. Main Concepts

Definition 6.
(refined neutrosophic matrix)
LetA=a11….a1m⋮⋱⋮an1…anm be an n×m matrix; if aij=x+yI1+zI2∈R2I, then it is called an refined neutrosophic matrix, where R2I is an refined neutrosophic field.Example 1.
X=I1I1+I23−I12I2 is a 2×2 refined neutrosophic matrix.Remark 2.
(addition and multiplication, ring structure)(a)
IfA is an m×n matrix, then it can be represented as an element of the refined neutrosophic ring of matrices such as the following: A=B+CI1+DI2, where D,B,andC are classical matrices with elements from ring R and from size m×n.For example,A=2+I1+3I21−I1−I23+4I21+I1=2131+1−101I1+3−140I2.(b)
The addition operation can be defined by using the representation in Remark2 as follows:(2)A+BI1+CI2+X+YI1+ZI2=A+X+B+YI1+C+ZI2.(c)
Multiplication can be defined by using the same representation as a special case of multiplication on refined neutrosophic rings as follows:(3)A+BI1+CI2X+YI1+ZI2=AX+AY+BX+BY+BZ+CYI1+AZ+CZ+CXI2.
This method of multiplication is exactly equivalent to the normal multiplication between matrices, but it is easier to deal with in this way.Example 2.
LetX=I1I1+I23−I12I2andY=−1I11+I23I1 be two refined neutrosophic matrices over the refined neutrosophic field of reals. We have(a)
X=A+BI1+CI2;A=0030,B=11−10,andC=0102.(b)
Y=M+NI1+SI2;M=−1010,N=0103,andS=0010.(c)
X+Y=−1+I12I1+I24−I1+I23I1+2I2.(d)
XY=−I1+I1+I21+I2I1I1+I1+I23I1−3+I1+2I21+I23−I1I1+2I23I1=I1+2I27I1−3+I1+4I28I1.(e)
If we compute the multiplication using the representation of Remark2, we get(4)AM=00−30,AN=0003,BM=0010,BN=040−1,BS=1000,CN=0306,AS=0000,CS=1020,CM=1020.
Hence,XY=AM+I1AN+BN+BM+BS+CN+I2AS+CS+CM=00−30+I11718+I22040=I1+2I27I1−3+I1+4I28I1.Theorem 3.
The set of all squaren×n refined neutrosophic matrices together makes a ring.Proof.
The proof holds directly from the definition ofn-refined neutrosophic rings by taking n=2.Remark 3.
The identity with respect to multiplication is the normal unitary matrix.Definition 7.
LetA be a square n×n refined neutrosophic matrix; then, it is called invertible if there exists a refined square n×n neutrosophic matrix B such that AB=Un×n, where Un×n is the unitary classical matrix.Theorem 5.
LetX=A+BI1+CI2 be a square n×n refined neutrosophic matrix; then, it is invertible if and only if A,A+C,andA+B+C are invertible. The inverse of X is X−1=A−1+A+B+C−1−A+C−1I1+A+C−1−A−1I2.Proof.
The proof holds as a special case of invertible elements in refined neutrosophic rings [8].Definition 8.
We define the determinant of a squaren×n refined neutrosophic matrix as detX=detA+detA+B+C−detA+CI1+detA+C−detAI2.
This definition is supported by the condition of invertibility.Theorem 6.
LetX=A+BI1+CI2 be a square n×n refined neutrosophic matrix; we have the following:(a)
X is invertible if and only if detX≠0(b)
IfY=M+NI1+SI2 is a square n×n refined neutrosophic matrix, then detXY=detXdetY(c)
detX−1=detX−1Proof.
(a)
IfdetX≠0, this will be equivalent to detA≠0,detA+C≠0,anddetA+B+C≠0, i.e., A, A+C,andA+B+C are invertible; thus, X is invertible according to Theorem 5.(b)
XY=AM+I1A+B+CM+N+S−A+CM+S+I2A+CM+S−AM. Hence, detXY=detAM+I1detA+B+CM+N+S+I2detA+CM+S  = detAdetM+I1detA+B+CdetM+N+S+I2detA+CdetM+S  = detA+I1detA+B+C−detA+C+I2detA+C−detAdetM+I1detM+N+S−detM+S+I2detM+S−detM=detXdetY.(c)
It holds directly from (b).Theorem 7.
LetX=A+BI1+CI2 be a square n×n refined neutrosophic matrix; we have the following:(a)
X is nilpotent if and only if A,A+C,andA+B+C are nilpotent(b)
X is idempotent if and only if A,A+C,andA+B+C are idempotentProof.
(a)
First of all, we will prove thatXr=Ar+I1A+B+Cr−A+Cr+I2A+Cr−Ar.We use the induction, forr=1, it is clear. Suppose that it is true for r=k, we prove it for k+1.(5)Xk+1=XkX=Ak+I1A+B+Ck−A+Ck+I2A+Ck−AkA+BI1+CI2=Ak+1+I1AkB+A+B+CkA+A+B+CkB+A+B+CkC−A+CkA−A+CkB−A+CkC+A+CkB−AkB+I2AkC+A+CkC−AkC+A+CkA−AkA=Ak+1+I1A+B+Ck+1−A+Ck+1+I2A+Ck+1−Ak+1.X is nilpotent if there is a positive integer r such that Xr=On×n. This is equivalent to(6)Ar=A+Cr=A+B+Cr=On×n,which implies the proof.(b)
The proof is similar to (a).Example 4.
Consider the following refined neutrosophic matrixA=2+I1+3I21−I1−I23+4I21+I1; we have the following:(a)
A=2+I1+3I21−I1−I23+4I21+I1=2131+1−101I1+3−140I2.B=2131, C=1−101, and D=3−140. B+D=5071,andB+C+D=6−172.(b)
B−1=−113−2,B+D−1=1/50−7/51,andB+C+D−1=2/191/19−7/196/19.(c)
A−1=B−1+I1B+C+D−1−B+D−1+I2B+D−1−B−1=−113−2+I1−9/951/1998/95−13/19+I26/5−1−22/53=−1−9/95I1+6/5I21+1/19I1−I23+98/95I1−22/5I2−2−13/19I1+3I2.It is easy to find thatA−1A=AA−1=1001.(d)
detB=−1,detB+D=5,detB+C+D=19,detA=−1+I119−5+I25−−1=−1+14I1+6I2.
If we compute the determinant ofA by using the classical way, we will get the same result.
Now, we illustrate an example to clarify the application of refined neutrosophic matrices in solving refined neutrosophic algebraic equations defined in [17].Example 5.
Consider the following system of refined neutrosophic linear equations:(7)2+I1+3I2X+1−I1−I2Y=−I1∗,3+4I2X+1+I1Y=I2∗∗.
The corresponding refined neutrosophic matrix isA=2+I1+3I21−I1−I23+4I21+I1.
SinceA is invertible, we get the solution of the previous system by computing the product:(8)A−1−I1I2=−1−995I1+65I21+119I1−I23+9895I1−225I2−2−1319I1+3I2−I1I2=I11+995−65+119I1−3−9895+225−1319+I2−2+3=−I11/19−6/19I1+I2.Thus,X=−119I1,Y=−619I1+I2.


## 4. Conclusion

In this paper, we have used the concept of refined neutrosophic set to build the corresponding refined neutrosophic matrix. On the contrary, many interesting properties have been studied and proved such as idempotency, nilpotency, determinants, and invertibility of these matrices.Also, a direct application of these matrices was proposed in solving refined neutrosophic equations.As a future research direction, we aim to study the diagonalization properties with eigenvectors of these matrices.



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