Inverse Method for Order n Matrix

 Corresponding author: Kasimu Juma Ahmed. Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution Liscense 4.0. Inverse Method for Order n Matrix Kasimu Juma Ahmed 1, *, Bashir Saidu Musa 2, Mustapha Muhammad Lamido 3, Shehu Adamu 4, Muhammad Bello Mustapha 5 and Ado Bappayo 6 1 Department of Mathematics and Statistics, Federal Polytechnic Bali, Taraba, Nigeria. 2 Department of Mathematical Sciences, Gombe State University, Nigeria. 3 Department of Mathematics and Statistics, Federal Polytechnic Bauchii, Nigeria. 4 Department of Computer Science, Federal Polytechnic Bali, Taraba, Nigeria. 5 Department of Science Laboratory Technology, Federal Polytechnic Bali, Taraba, Nigeria. 6 Department of Mathematics and Statistics, Federal Polytechnic Bauchii, Nigeria. World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 Publication history: Received on 12 April 2025; revised on 27 May 2025; accepted on 30 May 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.26.2.1978 Abstract The power of matrix algebra was seen not only in applied mathematics, applied sciences, engineering but also in economics, sociology, modern psychology and industrial management (i.e., system of linear equation, cryptography, optics, signal processing, image processing, graph theory, Machine Learning, Data Science etc.). In practice the matrix inverse methods is suitable only for non-singular small system because the higher the size of the system the more difficult finding the inverse of the system even with the help of software/application. With experience we were able to find the inverse of order 4, 5, … , n matrix with ease and also verified the method by computing 𝐴𝐴−1= 𝐼. Keywords: Matrix; Determinant; Inverse Method; Cryptography; Data Science 1. Introduction Matrix (Plural matrices) is a Square or rectangular array of an elements (which are usually numbers) in rows and columns. The general form of matrix with m rows and n column is: 𝐴= [ 𝑎11 𝑎21 ...𝑎1𝑛 𝑎12 𝑎22 ...𝑎2𝑛 ... 𝑎𝑚1 ... 𝑎𝑚2 ............ ...𝑎𝑚𝑛 ] Which is written in compact form as 𝐴𝑚𝑛= [𝑎𝑖𝑗]𝑚𝑛 1.1. Statement of the Problem In practice the matrix inverse methods is suitable only for non-singular small system (2 by 2 and 3 by 3 matrix). Hence, the need for order 4, 5, 6, … , n matrix. World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4149 1.2. Aim and Objectives of the Study This research paper aim to solve inverse of 4 by 4, 5 by 5, … , n b y n non-singular matrix. 1.2.1. The specific objectives are to • find the determinant of n by n matrix • test for non-singularity of the matrix if singular stop else • find 𝐴𝑑𝑗𝑜𝑖𝑛𝑡 (𝐴)= [𝑨𝒊𝒋]𝑇 • Compute 𝐴−1= 1 det (𝐴) ⨯𝐴𝑑𝑗𝑜𝑖𝑛𝑡(𝐴). • Verified by computing 𝐴𝐴−1= 𝐼. 1.3. Scope and Limitation The study is restricted in finding inverse of order 4, 5, … , n non-singular matrix. 1.4. Significance of the Study The field of matrix is fortunate to be blessed with lots of contribution but scholars used to restrict themselves on 3 by 3 matrix when it comes to matrix inverse method. As an optimizers we dimmed it fit to teach our student how to obtain inverse of order n because we do believe that unravelling the full strength of any method will certainly help our young one to think more deeply and be able to develop more sophisticated devices, applications etc. in time to come [1]. 1.5. Operational Definition of Basic Terms 1.5.1. Transpose is an operator that flips a matrix over its diagonal 1.5.2. Sign factor is a + (plus) or – (minus) sign that is attached to each entry element of a matrix (i.e., if (−1)𝑖+𝑗=𝑒𝑣𝑒𝑛 then the sign factor for that element 𝑎𝑖𝑗 is + else it must be – (odd)) 1.5.3 Minor (𝑴𝒊𝒋) is a determinant of some smaller square matrix generated from the original matrix (say A) by removing one or more of its rows and columns. 1.5.4 Cofactor (𝑨𝒊𝒋) it is calculated by multiplying the sign factor by the Minor 1.5.5 Adjoint matrix is the transpose of the cofactor matrix 1.5.6 Determinant is a scalar value computed for a given square matrix 2. History Matrix concept has ancient roots, with some early ideas found in Chinese mathematics. Over the years, matrices have seen an extended use in research, social science, commerce and are being used in cryptography, computer graphics, economics, chemistry, optics geology, animation, communication, wireless, signal processing, robotics, image processing, machine learning, data science and finance. Matrices also have in particular a wide range of applications in science and have been applied to solve real-world problems. Matrices are used to represent real-world data, message encryption and decryption, cryptography, coding theory, creating 3-D image and 2-D motion, to compress electronic data and to store fingerprint data, robotics and automation, CT scans and MRI scans, in economics to calculate gross domestic products, wireless application protocol, profit prediction, UV spectroscopy, automobiles, etc. The matrices are used in physics while applying Kirchhoff's Laws of Voltage and Current to solve problems, to explore electrical circuits, quantum mechanics and optics, to create graphs, calculate statistics, and conduct scientific research in a variety of domains. Matrices have a long history of use in solving linear equations, dating back to 300BC. [2, 3, 4, 5, 9, 10, 14]. 2.1. Overview: Inverse Method The square matrix A is called an invertible matrix if there exist a square matrix 𝐴−1 such that 𝐴𝐴−1=𝐼 (where I is a unit matrix, provided that the two matrices are of the same order). Then 𝐴−1 is called an invertible matrix of A, denoted by 𝐴−1. World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4150 It should be noted that, if 𝐴−1is a square matrix and det. (A) ≠ 0 then A is an invertible matrix, and we always have the property 𝐴𝐴−1= 𝐴−1𝐴=𝐼 and 𝐴−1= 1 𝑑𝑒𝑡 (𝐴) ⨯𝐴𝑑𝑗.(𝐴) [6, 7, 8, 11, 13, 12, 15]. The divisibility of determinants within square matrices serves as a captivating area of study, offering profound insights into the underlying structures and properties of these mathematical constructs. By exploring the divisibility of determinants, researchers delve into the intricate interplay between the elements within matrices, unraveling patterns and relationships that underpin their mathematical behavior. This pursuit extends beyond mere theoretical conjecture, finding practical relevance in various fields where matrices are indispensable tools for problem-solving and analysis. Understanding the divisibility of determinants empowers professionals to optimize their use of matrices in diverse applications, enhancing their efficacy in tasks ranging from data analysis to algorithm design. Some significant advancements have been made regarding the divisibility among determinants of power matrices. Moreover, the study of the divisibility of determinants within square matrices represents a testament to the enduring relevance and versatility of mathematical concepts across different domains of knowledge. As scholars probe deeper into this phenomenon, they uncover connections that transcend disciplinary boundaries, shedding light on the underlying principles governing complex systems and phenomena. An invertible matrix must be non-singular, meaning its determinant must be non-vanishing else the system is either linearly dependent or inconsistent. In Cryptography Matrix A is a key matrix which we used to encrypt our message and decrypt our messages by finding 𝐴−1. Hence, one of the importance of learning how to find inverse of large systems. In optimization we often pay attention to details (i.e., see how we can come out with a result that is more accurate and more convergent than the existing/common knowledge) [1, 3, 7, 15]. 3. Results 3.1. Linear Equation with two variables Write the system of equation 2𝑥+3𝑦=4 5𝑥+4𝑦=17 In matrix form, find 𝐴−1. Hence, solve the simultaneous linear equation Solution AX = B (1) [2 3 5 4][𝑥𝑦]= [4 17] Table 1 Cofactor of order 2 matrix (𝒂𝒊𝒋)th 𝒂𝒊𝒋 (−𝟏)𝒊+𝒋 𝑴𝒊𝒋 𝑨𝒊𝒋 𝑎11 2 + 4 4 𝑎12 3 - 5 -5 𝑎21 5 - 3 -3 𝑎22 4 + 2 2 |A| = det. (A) = ∑𝑎1𝑗 2 𝑗=1 𝐴1𝑗 = 𝑎11𝐴11+ 𝑎12𝐴12 (2) = 2(4) +3(-5) = -7 → non-singular, hence matrix A is invertible World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4151 𝐴−1= 1 |𝐴| ⨯𝐴𝑑𝑗.(𝐴) (3) Adj. (A) = (𝐴𝑖𝑗)𝑇 (4) Adj. (A) = [4 −3 −5 2] 𝐴−1= 1 −7[4 −3 −5 2] 𝑋=𝐴−1𝐵 (5) [𝑥𝑦] = 1 −7[4 −3 −5 2] [4 17] = [5 −2]. Therefore, (x, y) = (5, -2) 3.2. System of Linear Equation with three variables Write the system of equation in matrix form, find 𝐴−1, Hence, solve the simultaneous linear equation 𝑥−𝑦+3𝑧=5 4𝑥+2𝑦−𝑧=0 𝑥+3𝑦+𝑧=5 Solution AX=B [1 −1 3 4 2 −1 1 3 1][𝑥𝑦𝑧]= [5 0 5] Table 2 Cofactor of order 3 matrix (𝒂𝒊𝒋)𝒕𝒉 𝒂𝒊𝒋 (−𝟏)𝒊+𝒋 𝑴𝒊𝒋 𝑨𝒊𝒋 𝑎11 1 + 5 5 𝑎12 -1 - 5 --5 𝑎13 3 + 10 10 𝑎21 4 - -10 10 𝑎22 2 + -2 -2 𝑎23 -1 - 4 -4 𝑎31 1 + -5 -5 𝑎32 3 - -13 13 𝑎33 1 + 6 6 |A| = det. (A) = ∑𝑎1𝑗 3 𝑗=1 𝐴1𝑗 = 𝑎11𝐴11+ 𝑎12𝐴12+ 𝑎13𝐴13 (6) = 1(5) + (-1) (-5) + 3 (10) World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4152 = 40 → non-singular, hence matrix A is invertible 𝐴−1= 1 |𝐴| ⨯𝐴𝑑𝑗.(𝐴) (7) Adj. (A) = (𝐴𝑖𝑗)𝑇 (8) Adj. (A) =[5 −5 10 10 −2 −4 −5 13 6]𝑇 = [510 −5 −5 −2 13 10 −4 6] 𝐴−1= 1 40[510 −5 −5 −2 13 10 −4 6] 𝑋=𝐴−1𝐵 (9) [𝑥𝑦𝑧] = 1 40[510 −5 −5 −2 13 10 −4 6] [5 0 5] = 1 40[0 40 80] = [0 1 2]. Therefore, (x, y, z) = (0, 1, 2) 3.3. System of Linear Equation with four variables Write the system 𝑤+𝑥+𝑦−𝑧=2 4𝑤+4𝑥+𝑦+𝑧=0 𝑤−𝑥−𝑦+2𝑧=0 2𝑤+𝑥+2𝑦−2𝑧=2 In matrix form, find 𝐴−1. Hence, solve the simultaneous linear equation Solution AX=B Table 3 Cofactor of order 4 matrix (𝒂𝒊𝒋)𝒕𝒉 𝒂𝒊𝒋 (−𝟏)𝒊+𝒋 𝑴𝒊𝒋 𝑨𝒊𝒋 𝑎11 1 + -9 -9 𝑎12 1 - 2 --2 𝑎13 1 + 27 27 𝑎14 -1 - -17 17 𝑎21 4 - -1 1 𝑎22 4 + 0 0 𝑎23 1 - 3 -3 𝑎24 1 + -2 -2 𝑎31 1 - -2 -2 World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4153 𝑎32 -1 - 0 0 𝑎33 -1 + 5 5 𝑎34 2 - -3 3 𝑎41 2 - -3 3 𝑎42 2 + 1 1 𝑎43 2 - 10 -10 𝑎44 -2 + -6 -6 |A| = det. (A) =∑𝑎1𝑗 4 𝑗=1 𝐴1𝑗 = 𝑎11𝐴11+ 𝑎12𝐴12+ 𝑎13𝐴13+ 𝑎14𝐴14 (10) = 1(-9) +1 (-2) + 1 (27) + (-1)17 = -1 → non-singular, hence matrix A is invertible 𝐴−1= 1 |𝐴| ⨯𝐴𝑑𝑗.(𝐴) (11) Adj. (A) = (𝐴𝑖𝑗)𝑇 (12) Adj. (A) = [−9 1 −2 2 −2 0 0 1 27 −3 5 2 17 −2 3 −6]𝑇 = [−9 −2 27 17 1 0 −3 −2 −2 0 5 3 3 1 −10 −6] 𝐴−1= 1 −1[−9 −2 27 17 1 0 −3 −2 −2 0 5 3 3 1 −10 −6] 𝑋=𝐴−1𝐵 (13) [𝑤 𝑥𝑦𝑧]= 1 −1[−9 −2 27 17 1 0 −3 −2 −2 0 5 3 3 1 −10 −6][2 0 0 2]= [12 2 −34 −22]. Therefore, (w, x, y, z) = (12, 2,-34, -22) 3.4. System of Linear Equation with five variables Write the system of equation 2𝑣+3𝑤+𝑥+𝑦+4𝑧=34 5𝑣+2𝑤−4𝑥+2𝑦+𝑧=9 3𝑣+𝑤+2𝑥+3𝑦− 5𝑧=−19 𝑣+2𝑤+3𝑥+4𝑦+7𝑧=53 4𝑣−3𝑤+2𝑥−5𝑦+2𝑧=37 In matrix form, find 𝐴−1. Hence, solve the simultaneous linear equation World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4154 Solution AX=B [ 2 5 3 1 4 3 2 1 2 −3 1 −1 2 3 2 1 2 3 4 −5 4 1 −5 7 2 ] [ 𝑣 𝑤 𝑥𝑦𝑧 ] = [ 34 9 −19 53 37 ] Table 4 Cofactor of order 5 matrix (𝒂𝒊𝒋)𝒕𝒉 𝒂𝒊𝒋 (−𝟏)𝒊+𝒋 𝑴𝒊𝒋 𝑨𝒊𝒋 𝑎11 2 + 150 150 𝑎12 3 - 1650 --1650 𝑎13 1 + -516 -516 𝑎14 1 - -954 954 𝑎15 4 + 126 126 𝑎21 5 - 546 -546 𝑎22 2 + 444 444 𝑎23 -1 - -840 840 𝑎24 2 + -432 -432 𝑎25 1 - 162 -162 𝑎31 3 + -94 -94 𝑎32 1 - 202 -202 𝑎33 2 + -616 -616 𝑎34 3 - 54 -54 𝑎35 -5 + 366 366 𝑎41 1 - 4 -4 𝑎42 2 + 662 662 𝑎43 3 - 184 -184 𝑎44 4 + -594 -594 𝑎45 7 - 300 -300 𝑎51 4 + -248 -248 𝑎52 -3 - -256 256 𝑎53 2 + -284 -284 𝑎54 -5 - -252 252 𝑎55 2 + -60 -60 |A| = det. (A) =∑𝑎1𝑗 5 𝑗=1 𝐴1𝑗 = 𝑎11𝐴11+ 𝑎12𝐴12+ 𝑎13𝐴13+ 𝑎14𝐴14+ 𝑎15𝐴15 ………… (15) = 2(150) + 3 (-1650) + 1 (-516) + 1 (1954) + 4 (126) = -3708 → non-singular, hence matrix A is invertible World Journal of Advanced Research and Reviews, 2025, 26(02), 4148–4157 4155 𝐴−1= 1 |𝐴| ⨯𝐴𝑑𝑗.(𝐴) (16) Adj. (A) = (𝐴𝑖𝑗)𝑇 (17) Adj. (A) = [ 150 −546 −94 −4 −248 −1650 444 −202 662 256 −516 840 −616 −184 −284 954 −432 −54 −594 252 126 −162 360 −300 −60 ] 𝑇 𝑋=𝐴−1𝐵 (18) [ 𝑣 𝑤 𝑥𝑦𝑧 ] = 1 −3708 [ 150 −546 −94 −4 −248 −1650 444 −202 662 256 −516 840 −616 −184 −284 954 −432 −54 −594 252 126 −162 360 −300 −60 ] 𝑇 [ 34 9 −19 53 37 ] = [ 2 1 5 −2 6 ] Therefore, (v, w, x, y, z) = (2, 1, 5, -2 6) 4. Discussion Table 1, 2, 3 & 4 shows the 𝑎𝐼𝐽 entry elements of the given matrix, sign factor (−1)𝑖+𝑗 , minor (𝑀𝑖𝑗) and the respective cofactor (𝐴𝑖𝑗). Equation 1 shows the standard form of writing simultaneous linear equation in matrix form. Equation 2, 6, 10 and 15 present the formula for computing determinant of order 2, 3, 4 and 5 matrix respectively. Equation 3, 7, 11 and 16 present the formula for computing 𝐴−1 of order 2, 3, 4 and 5 matrix respectively Equation 4, 8, 12 and 17 present the formula for computing adjoint of order 2, 3, 4 and 5 matrix respectively. Equation 5, 9, 13 and 18 present the formula for computing unknowns of order 2, 3, 4 and 5 matrix respectively. It was proved that inverse method can be applied to non-singular matrix of higher order (i.e., 4, 5, . . . , n). 5. Conclusion Inverse method for solving order n non-singular matrix has been developed. Future research can consider implementing inverse method for solving order n matrix as software package and its deployment to different fields of knowledge. Compliance with ethical standards Disclosure of conflict of interest No conflict of interest to be disclosed. References [1] Ahmed, K. J. (2023). 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Property Norfolk State University, Norfolk, Virginia, USA Journal homepage: https://optimumscience.org Author’s short biography Ado Bappayo: Currently pursuing M.sc. Mathematics programme at Nasarawa State University Keffi, Nigeria. My recent publication was in “International Journal of Modelling and Applied Science Research. Manuscript ID: CJMA SR:2024-0055. Paper title: Prediction of Clustering Analysis Tool for Highway Traffic Network. Muhammad Bello Mustapha: Currently pursuing a master’s degree in Applied Ecology from the prestigious Abubakar Tafawa Balewa University, Bauchi, Nigeria. Some of my popularly known publications via Martaba FM Online (English) include the following; Nosocomial infections: Prevention and Universal precautions, Psychoactive effects of non-medical use of cannabis, Leagalization of cannabis (Marijuana) use on medical grounds, how to practice Biosafety Techniques, Intervention of Microorganisms to sustainable agriculture etc. Shehu Adamu: Currently pursuing Ph.D in Gazi University, Ankara, Turkiye by winning the prestigious scholarship of Turkiye Burslari. Main research mostly focusses in machine learning with application to educational data and agriculture and also have specialization in the role of IT in social development and less privilege environment.