Table 1 illustrates the title list of all journal articles that used Ajibade’s heart based method for rhotrix multiplication, available in the literature of rhotrix theory, starting from 2003 and to the end of 2013. Thus, articles in Table 1 belong to the class of commutative rhotrix theory. Now, we start a systemic review of these works in Table 1 as follows:

In [1] Ajibade introduced the concept of rhotrix of size three as

where is called the heart of any rhotrix. The operations of addition, scalar multiplication and multiplication are defined for rhotrices of size three in [1] . These rhotrix operations defined for rhotrix set of size three in [1] were thereafter, extended to rhotrix set of size in the Ph.D. thesis of Mohammed [5] , and recorded as follows: let

be the set of all real rhotrices of size, where, , is the integer value obtained

on division of by 2, and is the heart of any rhotrix in. Let and be

any two rhotrices in and scalar. Then

(4)

where and are the hearts of rhotrices and respectively.

The extended rhotrix multiplication (4) was named in [5] as “Ajibade’s heart-based method for multiplication of rhotrices”. The rhotrix operations defined in [1] was adopted by [6] to present various classifications of rhotrices and their expressions as abstract structures of groups, semigroups, monoids, rings and Boolean algebras. The theorem for rhotrix exponent rule was first proposed without proof in [6] , thereafter, [7] established and characterized the theorem for rhotrix exponent rule and extended the result to systemization of expressing special series and polynomial equations over rhotrices.

A remark on classifications of rhotrices as abstract structures was proposed by [8] over rhotrices. In the work, rhotrix ring was characterized; rhotrix integral domain and rhotrix field were constructed with certain conditions. Construction of certain field of fractions over rhotrices was presented by [9] as an extension to [8] . It was made known in [10] that the rhotrix field in [8] [9] holds only if the set of all hearty rhotrices of size three given in [6] is used as the underlying set.

The generalization of Ajibade’s heart based method for rhotrix multiplication in [5] was algorithmatized for computing machines by [11] . A simplification of rhotrix expression generalization in [5] was presented by [12] . Construction and analysis of metric topological spaces using rhotrix set as the underlying set were considered in [13] .

The concept of tree in graph theory was extended to rhotrix theory by [14] through their introduction of

rhomtrees of order as graphical representation of rhotrices of size, where. It

was shown in their work that these rhomtrees have connection to known real world models such as topology of computing network, methane compound and certain product of sets.

In [15] , the algebraic properties of singleton, coiled and modulo rhotrices were presented. Investigations of various constructions of finite fields over rhotrices were carried out in both [16] [17] . The cardinality of these finite fields was calculated through concrete examples. A study of the structure of rhotrices having entries from the set of integers modulo and their properties was conducted by [18] . The rhotrix quadratic polynomial presented as part of a note on rhotrix exponent rule and its applications in [7] was given certain extensions by [19] . Rhotrix polynomial and its extension to construction of rhotrix polynomial ring was proposed in [20] . An investigation of rhotrix sets and rhotrix spaces categorized over numbers in real and complex fields was presented by [21] . A system of linear equations arising from the rhotrix equation was investigated in [22] and the conditions for their solvability were determined in the article. A note on rhotrix system of equations was presented by [23] as an extension to earlier work considered in [22] . The system of rhotrix equations was solved simultaneously.

Table 2 illustrates the title list of all journal articles that used row-column based method for rhotrix multiplication, available in the literature of rhotrix theory from 2003 to 2013. Thus, articles in Table 2 belong to the class of non-commutative rhotrix theory. Now, we start a systemic review of works in Table 2 as follows:

Sani [3] proposed (5) as an alternative method for multiplication of rhotrices of size three as an attempt to answer the question of “how can one convert a rhotrix to matrix and then vice versa”, posed in the concluding

section in [1] .

This multiplication was later generalized by [4] to multiplication of rhotrices of size n as:

where and.

This (6) was presented in [5] as extended row-column based method for rhotrix multiplication.

In [24] , a presentation of the concept of Hilbert rhotrix and its relationship with well known Hilbert matrix was done. A special rhotrix termed as “Hilbert rhotrix” of size 5 was shown as a couple of two Hilbert matrices of sizes and.

A method of converting rhotrix to a special form of matrix called “coupled matrix” was given in [25] . This was achieved through rotating the rhotrix of size at an angle of 45˚ in anti-clockwise direction, which result into a special form of matrix with missing values. For example, a rhotrix of size 5 can express as a coupled matrix through half transpose as follows:

where indicates a rotation through 45˚ in anti-clockwise direction. The special matrix in (7) is a coupling of matrix with a matrix, hence, the name “coupled matrix”. Thus, in general,

This is a rhotrix expressed as a coupled matrix of dimension, coupling a matrix with a

matrix, where.

Two coupled matrices and can be multiplied together by simply filling the missing spaces with zeros, after the multiplication, we removed the zero in other to have the result in filled coupled matrix form. The following is a very useful result recorded from Sani [25] .

If a coupled matrix is completed with zeros, then its determinants is the product of the determinants of

the matrices and, where.

This result on coupled matrix is very significant because it can be applied to solve problems involving two different systems of linear equations simultaneously, where one is a system, while the other is a system,.

Sani [26] presented the solution of two coupled matrices by extending the idea of a coupled matrix in [25] to a general case involving and matrices.

A one-sided system of the form, where, is an n-dimensional rhotrix, the unknown n-dimensional rhotrix vector and the right hand side rhotrix vector was presented by [27] . The necessary and sufficient conditions for the solvability of the system of an n-dimensional rhotrix equation were discussed. Furthermore, the eigenvalues and the corresponding eigenvectors problems were solved.

The rhotrix addition and scalar multiplication defined in [1] was expressed in form of coupled matrices in [4] . The ideas were used by [28] to generalize and characterize the rhotrix vector space of size 3 initiated in [1] to rhotrix vector space of size n, through expression of rhotrices as coupled matrices.

Following this, [29] presented the concept of linear mapping to rhotrices and present its properties. It was shown in the work that the proposed method of converting a rhotrix to a “coupled matrix” (8) as defined in [25] is also a linear mapping.

In [30] an algorithm design for Sani’s row-column based method for rhotrix multiplication (6) was proposed. As an extension to [3] [4] [25] , various method of representing an arbitrary rhotrix was identified by [31] . One of the methods is the row-wise method, observed in the article to be flexible in analyzing rhotrices for mathematical enrichment. The flexibility of the representation has paved way for two formulae, one for row-column based method for arbitrary rhotrix multiplication and the other for heart-based method for arbitrary rhotrix multiplication.

The Cayley-Hamilton theorem for matrix is one of the well-known results in linear algebra. In 2012, the equivalence of this result was considered for rhotrix Cayley-Hamilton theorem in both [32] [33] . A note on relationship between invertible rhotrices and associated invertible matrices was proposed by [34] . A study of adjoint of a rhotrix and its basic properties was presented by [35] . The concept of inner product and bilinear forms over real rhotrices was considered in [36] . A determinant method for solving rhotrix system of linear equations was presented by [37] .

It is well known that an involutory matrix is a matrix that is its own inverse. Such matrices are of great im- portance in matrix theory and algebraic cryptography. In [38] a method for constructing involutory rhotrices and their properties was given. Thereafter, an extension to [38] was given by [39] through the development of some

theorems on involution in the context of rhotrices. Also, the description of Pascal rhotrices and their related properties was also considered. In [40] , the theory of graph was extended to consider adjacent rhotrix of a complete, simple and undirected graph. A consideration of parallel multiplication of rhotrices using systolic array architecture was presented by [41] . Thereafter, a rhotrix multiplication on two-dimensional process grid topologies was carried out by [42] . The concept of rhotrix linear transformation with a number of theorems was presented by [43] . An investigation of rhotrices and its elementary row operations was carried out in [44] .