Abstract The Use of Matrix Algebra in the Simplification of Accounting Transactions Based on the Principle of Double Entry Ajogbeje, Oke James Department of Mathematics, College of Education, Ikere Ekiti, Ekiti State, Nigeria Email: ajogbejeoke@yahoo.com There is need for students to be exposed to different methods of solving a given problem in order to facilitate a better understanding of accounting procedures based on the principle of double entry. This paper examined the use of matrix algebra in financial records. It is important to emphasize the use of matrix concepts for accountants, bankers and students of accounting to equip them for the emerging trends in the business/financial world. The business world is rapidly changing with new innovations in almost every discipline including accounting and finance. With the computer age, virtually all data and information required by management are going to be computerized before the turn of the decade. Hence, this calls for greater awareness in both students and practitioners of the accounting profession to have insight into the operations of the computer system. The aim of this paper is not to give a critical analysis of the computer operations but to bring into focus the use of matrix algebra in the simplification of some accounting procedures based on the principle of double entry. Keywords: Matrix Algebra, Financial Records, Accounting Transactions, Double Entry, Balance Sheet and Matrix Additive Introduction A matrix is a rectangular array of elements having m rows and n columns. Such array occurs in various branches of applied mathematics. In many cases, they form the coefficients of linear transformations or systems of linear equations arising for example from networks, frameworks in mechanics, curve fittings statistics and transportation assignment problems in business. Kreyszig (1988) opined that matrices enable us to consider all array of many numbers as a single object, denote it by a single symbol, and perform calculations with these symbols in a very compact form. Thus, the knowledge of how to manipulate matrix can be very useful to accounting practitioners in tackling problem that arise in business/ financial worlds. 351 www.hrmars.com/journals
Matrix Concept Lipschutz (1966), Children (1974), Micheal & Barry (1976), Oloidi (1997) and Ajogbeje, Durotoluwa, Abe, Oniyelu, Duromola & Adewoye (1998b) define matrix as a rectangular array of (real and complex) numbers. Thus, the general form of a matrix with m rows and n columns is We denote such a matrix by or simply and call it matrix. Note that the row and column of the element is indicated by its first and second subscripts respectively. As an illustration, the following matrices are examples: The matrices in (1) above are and If a matrix has only one row, we call it a row matrix or row vector. For example, Similarly, a column matrix or column vector is a matrix with only one column. For example, Addition and Subtraction of Matrices In this paper, only the addition and subtraction operations on matrices are needed in the manipulation of accounting records of interest while satisfying the basic principles of accounting. Matrix addition is defined only for matrices with the same shape (i.e. the same number of rows and the same number of columns). Hence, the addition of two matrices and written is the matrix obtained by adding corresponding entries or elements in A and B together: 352 www.hrmars.com/journals
For example, if and then Similarly, to subtract two matrices, the corresponding elements are subtracted in a similar process as above. That is, the entries in A B are obtained by subtracting corresponding entries in A and B. Application of Matrix Additive to Financial Records In the application of matrix addition concept, Hoel (1974), Freund (1975) & Ajogbeje et al (1998a) opined that one needs to examine thoroughly the accounting records to obtain the relevant figures expressed in monetary terms to form the required matrices. The principle of double entry in accounting provided the need to record any business transaction twice (i.e. as a debit and as a credit). The accounting records in the books are strictly guided by the principle of double entry, it is easier to obtain matrices of financial transactions from one period to another. Hence, using appropriate matrix concepts, the opening and closing balances can be ascertained from period to period without much problem. Consider the following transactions in Sea Way Nig. Ltd. for the year closing on 31 st December, 2010. i) Capital paid in N250 000 00 ii) Cash sales N200 000 00 iii) Credit sales N 30 000 00 iv) Cash purchase of stock N 100 000 00 iii) Credit purchase of stock N 50 000 00 iii) Cost of goods sold N 100 000 00 iii) Credit sales N 75 000 00 iii) Credit sales N 25 000 00 The Sea Way Nig. Ltd has the following accounts for recording each transaction entered into with customers. The accounts are coded with the assigned numbers or code numbers: 1 Capital; 2 Creditor; 3 Purchases; 4 Debtors; 5 Cash; 6 Sales; 7 Cost of Sales and 8 Expenses. According to the record of transaction, Sea Way Nig. Ltd has eight different accounts. Combining this fact together with the fact that the records are based on the principle of double entry, one needs to evolve a matrix. The rows represent the accounts to be 353 www.hrmars.com/journals
credited and the columns represent the corresponding accounts to be debited in order to satisfy the duality principle. The first transaction which is capital paid in requires a credit to the capital account and debit to the cash account that has received the amount. Thus account (1) is credited and account (5) is debited in the same amount of N 250 000 00. The second transaction required that cash account (5) is debited N 200 000 00 while sales account (6) is credited with the same amount of N200 000 00. This analysis is carried out until the last transaction is double classified and entered into the matrix format. The summary of the analysis is shown in the transaction matrix table below. Table 1: Summary of Account Transaction Entries Accounts Credited Accounts Debited 1 2 3 4 5 6 7 8 1 250 000 00 2 50 000 00 3 10 000 00 4 75 000 00 5 100 000 00 25 000 00 6 30 000 00 20 000 00 7 8 The transaction matrix above shows the flow of wealth between two balance sheet data. The problem of evolving a trial balance is removed by the simultaneous entry to credit and debit sides. In the above table, each row represents all credit entries to a particular account. Similarly, each column depicts all the debit entries to a particular account. Thus the balance in each account can determine accurately by obtaining the difference between the sum of each credit and each debit as recorded in each row and column. For example, the balance for, say, the creditor s account is given by the sum of the credit in the creditor s account, that is, This procedure is repeated for each of the accounts to ascertain the balances. Let this balances obtained for Capital, Creditor, Purchases, Debtors, Cash, Sales, Cost of Sales and Expenses be and respectively. These balances obtained from the transaction matrix can be combined together to form a transaction vector, that is, 354 www.hrmars.com/journals
In obtaining the value for each of the balances and, consideration was given to the accounting principle that assets plus expenses must be equal to the liabilities incurred plus the revenue generated by the business (Lewis & Gillespie, 1982; Bull, 1984). Expressed in mathematical terms, we have This equation explains why we need to subtract the sum of credit entry from the sum of the debit entry to obtain the excess of credits. The transaction vector can be further reduced using accounting concept by combining those that are revenue and expenses to give the excess over revenue, which is profit. This defined by the profit sub vector that is, The new reduced transition vector is given by The reduced transition vector is the balance sheet as at 31 st December, 2010 and this also becomes the opening balance for the following year (January 1, 2011) business period. To verify the veracity of the above assertion, the accounting records of the above transaction is hereby presented using the conventional method as proof according to their code number. Sea Way Nig. Ltd Ledger Accounts Capital Account (1) Creditors Account (2) Cash 250 000 00 Purchases Account (3) Purchases 50 000 00 355 www.hrmars.com/journals
Cash 100 000 00 Creditors 50 000 00 150 000 00 Balance b/d 50 000 00 Debtors Account (4) Sales 30 000 00 Balance c/d 45 000 00 75 000 00 Cost of goods sold 100 000 00 Balance c/d 50 000 00 150 000 00 Cash 75 000 00 75 000 00 Balance b/d 45 000 00 Cash Account (5) Capital 250 000 00 Sales 200 000 00 Debtors 75 000 00 525 000 00 Balance b/d 400 000 00 Sales Account (6) Balance c/d 230 000 00 230 000 00 Cost of Goods Sold Account (7) Purchases 100 000 00 Miscellaneous Expenses 25 000 00 Balance c/d 400 000 00 525 000 00 Cash 200 000 00 Debtors 30 000 00 230 000 00 Balance b/d 230 000 00 Purchases 100 000 00 Miscellaneous Expenses Account (8) Cash 25 000 00 Figure 1: Sea Way Nig. Ltd Ledger Accounts 356 www.hrmars.com/journals
Sea Way Nig. Ltd Trial Balance as at 31 st December, 2010 Capital ( ) Creditors ( ) Purchases ( ) Credit Balance on Debtors Account ( ) Cash ( ) Sales ( ) Cost of Goods Sold ( ) Miscellaneous Expenses ( ) DR 50 000 00 400 000 00 100 000 00 25 000 00 575 000 00 CR 250 000 00 50 000 00 45 000 00 230 000 00 575 000 00 Figure II: Sea Way Nig. Ltd Trial Balance Sea Way Nig. Ltd Trading, Profit and Loss Account for the Year Ended 31 st December, 2010 # Sales ( ) Less: Cost of Sales ( ) Gross Profit Less: Miscellaneous Expenses ( ) Net Profit ( ) Figure III: Sea Way Nig. Ltd Trading, Profit and Loss Account 230 000 00 100 000 00 130 000 00 25 000 00 105 000 00 Sea Way Nig. Ltd Balance Sheet as at 31 st December, 2010 Stock ( ) 50 000 00 Cash ( ) 400 000 00 450 000 00 Less: Creditors falling due within one year ( ) 95 000 00 Net Asset 355 000 00 Financed by: Capital ( ) 250 000 00 Undistributed Profit & Loss ( ) 105 000 00 355 000 00 Figure IV: Sea Way Nig. Ltd Balance Sheet 357 www.hrmars.com/journals
From the above comparison, it is ascertained that Fig. I represents the Sea Way Nig. Ltd Ledger Accounts. Fig. II denotes the Trial Balance of the same company. Fig. III represents the Trading, Profit and Loss Account of the business while Fig. IV depicts the Balance Sheet of the company. Aside this, the above analysis and comparison also established the fact that matrices or vector algebra could be used in double entry accounting in all stages of financial statements. Conclusion The matrix method of obtaining opening and closing balances for any accounting period is very efficient, accurate and less time consuming. The method is likely to gain acceptance by practicing accountants, accounting educators and student in post primary and post secondary institutions. Recommendations Based on the above result, it is recommended that the matrix method of obtaining opening and closing balances for any accounting period should be taught to the accounting and finance students in our post primary and post secondary institutions alongside the conventional method being presently used. This will enable our students to appreciate the beauty and application of financial or business mathematics in accounting and finance related disciplines. Acknowledgement I wish to acknowledge the support and encouragement received from the Management Sea Way Nigeria Ltd, Owo Ondo State, Nigeria during the data collection for this study. References Ajogbeje, O. J., Oloidi, G. A., Adeyeye, P. O., Abe, T. O. & Kolawole, F. O. (1998). Business Mathematics [Volume I]. Akure: STECOM Publishers. Ajogbeje, O. J., Durotoluwa, J. O., Abe, T. O, Oniyelu, S. O., Duromola, M. K. & Adewoye, R. A. (1998). Algebra Volume I. Akure: STECOM Publishers. Bull, R. J. (1984). Accounting in Business, 5 th Edition. Butterworths: English Language Book Society. Children, R. L. (1974). Mathematics for Managerial Decisions. New Jersey: Prentice Hall Inc Freund, J. E. (1975). College Mathematics with Business Application, London: Prentice Hall Inc. Hoel, P. G. (1974). Finite Mathematics and Calculus with Application to Business. New Jersey: John Wiley & Sons Inc. Lewis, A. & Gillespie, J. (1982). Foundation in Accounting. London: Prentice Hall Inc. 358 www.hrmars.com/journals
Lipschutz, S. (1966). Theory and Problems of Finite Mathematics. McGraw Hill Book Co. Micheal, G. & Barry, S. (1976). Mathematics for Financial Analysis. Oxford: Schaum s Series. Oloidi, G. A. (1997). Business Mathematics. Akure: Fajimi Publishers Ltd. 359 www.hrmars.com/journals