The research work is solely aimed at solving ystem of linear equation is a different way. System of solving linear equation may result into rectangular Augmented matrix or square Augmented matrix. The methods used this research work has led to a new way or technique in solving system of linear equation – be it in the rectangular or square Augmented matrix form. Most pairs of simultaneous linear equations are usually represented by square augmented matrices with unknown variables, usually 2 x 2 and 3 x 3 matrices.
From this research work, it is now possible to:
a. Solve simultaneous equations arising from rectangular Augmented matrix of 3 x 2 and 4 x2 order
b. Find the determinant of 2 x 2 and 3 x 3 with a different method – that never existed before.
c. Solve 2 x 2 and 3 x 3 and 4 x 4 aquare Augmented matrix with a new method using determinant but quite different from that of crammer’s rule or method.
Table of Contents
1. INTRODUCTION
2. EHIMETALOR’S FORMULA FOR SOLVING 3 X 2 AUGMENTED MATRIX WITH THREE UNKNOWN VARIABLES.
3. SIMULTAENOUS LINEAR EQUATIONS (SLE) INVOLVING TWO UNKNOWN VARIABLES
4. SOLVING SIMULTANEOUS LINEAR EQUATIONS (SLE) INVOLVING SQUARE AUGMENTED MATRIX (3 X 3 MATRIX)
5. DETERMINANT OF 3 x 3 MATRIX
6. CONCLUSION
Research Objectives and Key Topics
The primary research objective is to develop and introduce a novel mathematical technique for solving systems of simultaneous linear equations, specifically addressing both rectangular and square augmented matrix forms, which traditional methods like Cramer's rule or Gaussian elimination may struggle with or fail to solve efficiently.
- Mathematical derivation of Ehimetalor’s formula for rectangular and square augmented matrices.
- Development of a new method for calculating the determinant of 2x2 and 3x3 matrices.
- Methodological application for solving systems with multiple unknown variables.
- Comparative analysis between classical methods (Cramer, Gauss, Sarrus) and the proposed approach.
- Practical demonstration of the new methodology through worked examples and structured assessment problems.
Excerpt from the Book
1. INTRODUCTION
It is quite difficult to solve Simultaneous linear equations with rectangular augmented matrix such as 2 x 3 matrix, 3 x 2 matrix, 2 x 4 matrix, 4 x 2 matrix, 3 x 4 matrix, 4 x 3 matrix, etc. with unknown variables.
Each problems is solved using different methods which is quite different from the solution of the other forms of equations.
In this research work, I was able to solve the pair of Simultaneous linear equation of rectangular augmented matrix of the following orders: 3 x 2 and 4 x 2 matrix, giving rise to new formulas.
Summary of Chapters
1. INTRODUCTION: Outlines the difficulty of solving specific rectangular augmented matrix systems and introduces the motivation for developing new formulas.
2. EHIMETALOR’S FORMULA FOR SOLVING 3 X 2 AUGMENTED MATRIX WITH THREE UNKNOWN VARIABLES.: Details the algebraic derivation of the new formula specifically for 3x2 matrices using elimination techniques.
3. SIMULTAENOUS LINEAR EQUATIONS (SLE) INVOLVING TWO UNKNOWN VARIABLES: Provides the methodologies and formulas for solving systems with two variables using the proposed approach and traditional matrix comparisons.
4. SOLVING SIMULTANEOUS LINEAR EQUATIONS (SLE) INVOLVING SQUARE AUGMENTED MATRIX (3 X 3 MATRIX): Extends the new method to 3x3 square augmented matrices with worked step-by-step examples.
5. DETERMINANT OF 3 x 3 MATRIX: Focuses on the calculation of determinants for 3x3 matrices using the Ehimetalor method compared against classic Sarrus and Cramer methods.
6. CONCLUSION: Summarizes the achievements of the research, successes in specific matrix dimensions, and acknowledges existing limitations for higher-order systems.
Keywords
Simultaneous equation, Simultaneous linear equation, Rectangular Augmented Matrix, Linear Equation, Equations, Unknown variables, Solving Equations, Solving Linear Equations, Square Augmented Matrix, Determinant, Ehimetalor’s formula, Elimination method, Substitution method, Matrix method.
Frequently Asked Questions
What is the core focus of this research paper?
The research paper focuses on developing a new algebraic technique to solve systems of simultaneous linear equations, specifically targeting forms like rectangular and square augmented matrices.
What are the central themes of the work?
The work centers on matrix algebra, specifically deriving new formulas for matrix systems that are typically challenging for standard methods like Cramer’s rule.
What is the primary goal of this research?
The aim is to provide a reliable method to solve rectangular and square augmented matrices with multiple unknown variables, providing a set of formulas that serve as an alternative to conventional linear algebra techniques.
Which scientific methods are employed?
The author uses algebraic elimination and substitution methods combined with determinant theory to derive "Ehimetalor’s formula" for specific matrix orders.
What topics are discussed in the main sections?
The main sections cover 3x2 rectangular systems, 2-variable SLEs, 3x3 square augmented systems, and the novel computation of matrix determinants.
Which keywords define this research?
Key terms include Simultaneous Equations, Rectangular Augmented Matrix, Square Augmented Matrix, Determinant, and Ehimetalor’s Formula.
How does Ehimetalor’s formula differ from Cramer's rule?
While both rely on determinant-based concepts, the author presents the new formula as a distinct technique that is applied differently to rectangular matrices where traditional methods like Cramer's rule are limited.
What limitation does the author acknowledge regarding rectangular matrices?
The author notes that while 3x2 and 4x2 systems are solved, larger complex orders like 2x3, 3x4, or 5x5 matrices remain challenges for future volumes of this research.
Can this method be applied to 4x4 systems?
Yes, the research includes methodology for solving 4x4 square augmented matrices using determinants with a distinct calculation process.
- Quote paper
- H. Ehimetalor (Author), 2023, Simultaneous Linear Equations Journal. New Method of Solving, Munich, GRIN Verlag, https://www.grin.com/document/1380026
