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

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]. %e concept of refined neutrosophic structure was supposed firstly in [6] by splitting indeterminacy I into two levels of subindeterminacies I1 and I2. %is 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]. %rough 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 field F(I1, I2). %e structure of refined neutrosophic numbers is taken as a + bI1 + cI2 instead of (a, bI1, cI2). %is 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.


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]. e concept of refined neutrosophic structure was supposed firstly in [6] by splitting indeterminacy I into two levels of subindeterminacies I 1 and I 2 . is idea was used in the study of refined neutrosophic rings [7][8][9], modules [10,11], and groups [6]. Recently, the concept of n-refined neutrosophic structures was defined and used in [12][13][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].
rough 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 field F(I 1 , I 2 ). e structure of refined neutrosophic numbers is taken as a + bI 1 + cI 2 instead of (a, bI 1 , cI 2 ). is 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.

Preliminaries
Definition 1 (see [7]). Let K be a field, the neutrosophic field generated by K ∪ I, which is denoted by K(I) � K ∪ I.
Definition 2 (see [7]). Classical neutrosophic number has the form a + bI, where a and b are real or complex numbers and I is the indeterminacy such that 0 · I � 0 and I 2 � I which results in I n � I for all positive integers n.
Definition 3 (neutrosophic matrix; see [15]). Let M m×n � ( a ij ): a ij ∈ K(I) , where K(I) is a neutrosophic field. We refer this to be the neutrosophic matrix.
Remark 1 (see [6]). e element I can be split into two indeterminacies I 1 and I 2 with conditions Definition 4 (see [1]). If X is a set, then X(I 1 , I 2 ) � (a, bI 1 , cI 2 ): a, b, c ∈ X is called the refined neutrosophic set generated by X, I 1 , and I 2 .
It is called a neutrosophic field if R is a classical field.
Theorem 2 (see [17]). Let A 1 X 1 + · · · + A n X n � C, , be a linear equation with n-variables over a refined neutrosophic field F(I 1 , I 2 ). en, it is equivalent to the following system of classical linear equations over the classical field F:

Main Concepts
Definition 6 (refined neutrosophic matrix) then it is called an refined neutrosophic matrix, where R 2 (I) is an refined neutrosophic field.
Remark 2 (addition and multiplication, ring structure) (a) If A is an m × n matrix, then it can be represented as an element of the refined neutrosophic ring of matrices such as the following: where D, B, and C are classical matrices with elements from ring R and from size m × n.
(b) e addition operation can be defined by using the representation in Remark 2 as follows: (c) Multiplication can be defined by using the same representation as a special case of multiplication on refined neutrosophic rings as follows: is method of multiplication is exactly equivalent to the normal multiplication between matrices, but it is easier to deal with in this way.
be two refined neutrosophic matrices over the refined neutrosophic field of reals. We have (e) If we compute the multiplication using the representation of Remark 2, we get 2 Journal of Mathematics Hence, XY � AM + I 1 (AN + BN + BM + BS + CN)

Theorem 3.
e set of all square n × n refined neutrosophic matrices together makes a ring. Proof.
e proof holds directly from the definition of nrefined neutrosophic rings by taking n � 2.

Remark 3.
e identity with respect to multiplication is the normal unitary matrix.
Definition 7. Let A 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 � U n×n , where U n×n is the unitary classical matrix.

Theorem 5. Let X � A + BI 1 + CI 2 be a square n × n refined neutrosophic matrix; then, it is invertible if and only if A, A + C, and A + B + C are invertible. e inverse of X is
Proof.
e proof holds as a special case of invertible elements in refined neutrosophic rings [8].
□ Definition 8. We define the determinant of a square n × n refined neutrosophic matrix as detX � detA is definition is supported by the condition of invertibility. Theorem 6. Let X � A + BI 1 + CI 2 be a square n × n refined neutrosophic matrix; we have the following:  We use the induction, for r � 1, it is clear. Suppose that it is true for r � k, we prove it for k + 1.

Journal of Mathematics
X is nilpotent if there is a positive integer r such that X r � O n×n . is is equivalent to which implies the proof.
(b) e proof is similar to (a).
□ Example 4. Consider the following refined neutrosophic matrix A � 2 + I 1 + 3I 2 1 − I 1 − I 2 3 + 4I 2 1 + I 1 ; we have the following: It is easy to find that If we compute the determinant of A 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: e corresponding refined neutrosophic matrix is Since A is invertible, we get the solution of the previous system by computing the product:

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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest. 4 Journal of Mathematics