Natural rhotrix

Abstract: This paper takes a look at a set of rhotrix whose entries are ordered natural numbers. This rhotrix is called the natural rhotrix. Properties of this set are examined and the results are presented. Since the natural rhotrix R is not invertible, a maiden investigation is made into the concepts of minor rhotrices of R, determinant functions (h(R)), codeterminant function (codet(R)), and index ( ) of a natural rhotrix. It was found that codet(R) = h(R), and their methods of computations are outlined for mathematical enrichment.


PUBLIC INTEREST STATEMENT
A natural rhotrix is a rhotrix whose elements are all countable numbers. A rhotrix is a system of numbers in a rhomboid shape. They are similar to matrices. These systems of numbers form sets of rhotrices depending on the amount of numbers in the systems. Each set obeys some mathematical rules depending on the type of numbers you have in the set and behave differently.
This work examines the behavior of a rhotrix set whose elements are countable numbers. In real life, children from different families behave differently. Therefore, a scientific study of each family will improve inter human relationship and help to harness the talents inherent in each of the family. This is to illustrate the essence of the scientific study of families or sets of physical properties around us like this work.
Detail introduction, transformation into matrices and applications abound in literature.
in developing and expanding this concept, most often, in analogy with the concepts of matrices usually through a transformation that converts a matrix into a rhotrix and vice versa (Ajibade, 2003;Sani, 2008). One of such works was the classification of rhotrices into sets and algebraic spaces by Mohammed and Tella (2012). The paper classifies rhotrices into natural rhotrix set, real rhotrix set, complex rhotrix set, rational and irrational rhotrix sets. Thus, their work has actually opened up different branches of studying rhotrices. Therefore, this article is picking on the first branch, the natural rhotrix set. Furthermore, in this paper, most of these properties of natural rhotrices will be examined without necessarily having to go through a transformation. Though, if need be, these properties would always be in conformity to one transformation or the other. This can be examined.
Definition 1.1 Mohammed and Tella (2012) A rhotrix set is called a natural rhotrix set if its rhotrice entries belong to the set of natural numbers. For example, is the set of all three-dimensional natural rhotrices.
This set of natural rhotrix is a beautiful rhotrix with unique characteristics many of which are yet to be discovered.
In this work, we will be looking at some fundamental properties of this algebraic set.
Definition 1.2 Ajibade (2003), Mohammed and Tella (2012) A real rhotrix set of dimension three, denoted as R 3 (ℜ) was defined by Ajibade as where h(R) = c is called the heart of any rhotrix R belonging to R 3 (ℜ) and ℜ is the set of real numbers. Examples showing extension of this set and analysis are copious in literature (Aminu & Michael, 2015;Baumslag & Chandler, 1968;Ezugwu, Ajibade, & Mohammed, 2011;Mohammed, 2009Mohammed, , 2014Mohammed, Balarabe, & Imam, 2012;Tudunkaya & Manjuola, 2010;Usaini & Mohmmed, 2012). It is worthy to note that an n-dimensional real heart-based rhotrix denoted by R n (ℜ), will have it cardinality as |R n (ℜ)| = 1 2 (n 2 + 1), where n ∈ 2ℤ + + 1. This implies that all heart-based or heart-oriented rhotrices are of odd dimension (≥3). All operations (addition and multiplication) in this work, will be as defined by Ajibade in (2003). Thus, addition and multiplication of two heart-based rhotrices are defined as: and respectively. A generalization of this hearty multiplication is given in Mohammed (2014) and in Ezegwu et al. (2011), ⟩ A row-column multiplication of heart-based rhotrices was proposed by Sani (2004) as: A generalization of this row-column multiplication was also later given by Sani (2007) as: where R n and Q n are n-dimensional rhotrices (with n rows and n columns).

Definition 1.3 A rhotrix is said to be invertible if it has an inverse
For example let Then, the inverse of A is given as It is observed that any rhotrix h(R) ≠ 0 is an invertible or non-singular rhotrix. However, a natural rhotrix is a singular rhotrix. That is, we cannot find A −1 for which A is a natural rhotrix.

Preliminaries
In this section, we shall strictly concern ourselves with the natural rhotrix-a rhotrix whose parent set is the set of well-ordered natural numbers. A natural rhotrix starts with a dimension one (i.e. R 1 ). Therefore, the cardinality of n-dimensional natural rhotrix is given by |R n (ℕ)| = 1 2 (n 2 + 1), where n ∈ 2ℕ + 1.
Definition 2.1 (Major row and major column) The major row and the major column are usually the only full row and full column in a rhotrix. They are usually at the middle of the rows and columns of any dimensional rhotrix.
A natural rhotrix, as other rhotrices, has one major row and one major column.

Examples of natural rhotrices
This section gives different representations of natural rhotrices as examples according to their dimensions.
(a) A natural rhotrix of dimension one (R 1 ) is given by: where a ∈ ℕ. It only has a heart with no other entries.
Remark 3.2 You can go on and on. For example, R 9 and R 11 will have their last entries as 41 and 61, respectively.

Remark 3.3
The above is the generalization of any natural rhotrix.

Properties of natural rhotrix
Lemma 4.1 Let R i be any i dimensional natural rhotrix. Then, the heart (h(R i )) is the middle value of a set of n numbers that make up the rhotrix if and only if n = |R i | i = 1, 3, 5, … Proof First part Since n ∈ 2ℕ + 1, then there exist meddle value (median).
The converse follows from the Cardinality of R n where n ∈ 2ℕ + 1.

Determinant function
Though, the natural rhotrices are not invertible rhotrices. However, the heart of a natural rhotrix plays the role of a determinant function.

Lemma 4.2 Let A and B be any natural rhotrices of dimension n and |A| a determinant function of A , then
Proof Let |A| = h(A) and |B| = h(B) then

Remark 4.2
(i) Again, the justaposition AB represents A•B as defined in Ajibade (2003) (ii) We will adopt h(A) to mean the determinant function of A.

The determinant function of the above rhotrix A is designated by h(A)
Computing the value of the determinant function of any natural rhotrix is simply the value of its heart. However, for higher natural rhotrices, we introduce the concept of minor rhotrices and codeterminant functions. That brings us to the next subsection.

Codeterminant function
The concept of codeterminant functions is similar to that of minor matrices. In natural rhotrices of dimension three (R 3 ), the codeterminant function is the same as the determinant function. But for higher natural rhotrices the codeterminant functions are not necessarily the same. To evaluate the codeterminant function of a higher natural rhotrix, we need to first split or reduce the higher natural rhotrix into chain(s) of R 3 called the minor rhotrices and the determinant function of each of them is evaluated as above. These minor rhotrices are split or reduced either along the major column or along the major row and their determinant functions are summed up accordingly. For a natural rhotrix with well-ordered entries, the result is the same regardless of whether it is summed up along the major column or along the major row.
Consider the natural rhotrix of dimension five below: To evaluate codet(A), the higher rhotrix needs to be split into minor rhotrices of dimension three.
That is splitting the rhotrix A along the major column gives: Next, we find codeterminant function, first along the major column gives: Now along the major row gives: Extension to higher dimension can be made in a similar manner used in reducing R 5 to minor rhotrices of R 3 . Let us consider the next example.
It is of utmost importance that the reader practices evaluating the codeterminant functions of higher rhotrices along major row or major column. This will give the reader the prerequisite confidence in obtaining the values of codeterminant functions in certain cases where a particular major row or column tends to accelerate the release of results which consequently leads to reduced hardship and computation time expended. Considering the Example 4.2, summing along the column seems to make life easier for the reader. However, the beauty of this work lies in its simplicity.

Index of rhotrices
The index of a natural rhotrix A is the number of minor rhotrices of dimension three that can be derived, either along the major column or along the major row, from A. This index is a whole number or better still a natural number. For example the index of R 3 is 1 and of R 5 , R 7 , and R 9 are 2, 3, and 4, respectively. Appropriately, the index of R 1 is zero.
Theorem 4.2 Given any rhotrix R, the codet(R) = h(R) where is a natural number called the index of R.
Proof We prove using mathematical induction. Since an index of a natural rhotrix is a natural number corresponding to the number of R 3 that can be derived from R n , n ≥ 3, and n ∈ 2ℕ + 1. Now, when n = 3, the since the codet(R 3 ) is necessarily the det(R 3 ) = h(R 3 ). By Lemma 4.1. Implies that = 1, So, the equation is true for n = 3.
For n = 5, then we have two minors of R 3 . That is, implies that = 2. So, the equation is true for n = 5.
For n = 7, then, we have three minors of R 3 , then So, the equation is true for n = 7 and = 3.

Theorem 4.3 Giving any natural rhotrix R
where is the index and |R| is the cardinality of R, and |R n | = 1 2 (n 2 + 1) Proof Since codet(R) = h(R) and by Lemma 4.1, determinant function is h(R). Then, the result follows from the Theorem 4.1.

Conclusion
This article examined the properties of the natural rhotrix set, introduced the concepts of minor rhotrices, determinant functions, codeterminant functions and index of natural rhotrices. These concepts are a novelty to rhotrix algebra, and their methods of computations are presented for mathematical enrichment. Using the results in this paper, one can evaluate the determinant function of any n-dimensional rhotrix at a glance. With the links between the determinant functions, codeterminant functions and the index one will be able do a sketch of any n-dimensional natural rhotrix no matter how large is the value of n.