Data-Pattern Index

Matrix Accounting Description

Matrix Accounting Provides New Audit Tools and Additional Information

Recent improvements in matrix accounting are expected to add a number of forensic attributes to routine accounting and audit practice. The new tools address the two audit risks of possible non-neutral financial reporting and possible unusual entries which conceal a loss. Matrix accounting introduces a new area of thought, analytical review of internal control, and provides new information which facilitates better management of the accounting process.

Matrix accounting effectively posts complete journal entries, rather than ledgers, with every general journal entry identified by its combination of account-sides (Figure 1). Transactions are identified by their content. Entries with identical account-sides are accumulated (posted) to a summary. Each summary contains a list of its component entries, thereby enabling 100% classification and search of general journal entries.

The summaries in matrix accounting substitute for the ledgers of conventional accounting. In this new organization, conventional ledger totals may be obtained via indexes, but the relationship of accounts within the transaction remains intact, i.e., the "other ends" of a transaction are always visible. While matrix organization appears to be neutral for financial accounting, it is positive for managerial accounting and auditing.

New Audit Tools

Only a small fraction of frauds is found in the books of entry. Managerial frauds, from WorldCom to Rentway, have evaded auditors. Account-side and range of account-side (a) (listed below) help to detect managerial frauds. Employee frauds often leave unusual combinations that are obvious to the auditor or supervisor. Unusual entries may often "pop out" in display of an account-side. Combined searches (listed below) are also useful in finding unusual entries.

Partial List of Audit Searches in Matrix Accounting

By account-side Aids in judgment for internal control, supervision, and aggregation. The number of summaries may indicate a wide or narrow range of activities affecting the account. A greater count of summaries is more likely to require compensating controls.

By transaction frequency Separates the repetitive from the exception and facilitates supervision of amortization entries.

By complexity Separates simple transactions from more complex ones. Payroll, cost accounting and spreadsheet entries may be eliminated or selected for further scrutiny by specifying fewer than x accounts or greater than x accounts.

Specific Boolean Useful when following a specific question. Two or more account- sides are entered and the search returns all entries containing all the named account-sides.

Combined searches Useful for finding specific types of transactions or accounts; for example, in the receivables area, a search for debit to the allowance account and a minimum of three accounts or more to find partial charge-offs; or a range selection of several allowance accounts (see below) and subsequent recapitulation to find all related accounts in the range of "ledger" accounts searched.

Ranges of account-side (a) (set theory operations on range selection) Most firms organize ledger accounts in ranges by financial report sector, i.e., property accounts or maintenance accounts. In these situations, one can select for one list all entries containing debits to fixed assets, and in a second list, all entries containing credits to maintenance expense. Then, a set search for the intersection of the two lists will identify all summaries containing at least one fixed asset debit and at least one maintenance expense credit. While some audit software does perform this specific search, matrix accounting supports mixed-set searches over the entire trial balance. Note that a zero result provides negative assurance.

Special Conditions

The following journal entry conditions are not part of matrix accounting, but knowledge of transaction content can help in supervision and audit. Perhaps individual firms will add their ownspecial conditions . These conditions may be applied to summaries or to the component entries of a summary:

Traffic Report

The information in the ledger is the same as in journals. Assuming that special journals are included as totals in the general journal, a matrix accounting report should reconcile completely with existing records. Only one attribute is new to financial reporting in matrix organization: the count of journal entry summaries posting each and every ledger account. That attribute adds considerable supervision and auditing ability. A proposed traffic report is shown in Figure 2.

Effect of Spreadsheets

The frequent practice of using spreadsheets to accumulate batches of transactions for journal entry reduces the direct identification of transactions. Matrix accounting nevertheless identifies un-batched transactions. Searches for long entries (up to 9,999 lines) are expected to organize spreadsheet entries. The selection of entries posting a specific account-side often identifies inconsistencies in spreadsheet usage. The proposed near-neighbor search would identify minor variations in complex entries.

Downloading of selected summaries to spreadsheets is useful for further reporting. Identifying which underlying journal entries need further scrutiny may depend on local systems.

Analytical Review of Internal Control

The transaction patterns for each account (as illustrated in Figures 1 and 2) can quickly guide the auditor to search for unusual transactions. Similarly, the count of supporting entries to a summary provides evidence of under- or over-amortization. Range searches by specific divisions (see above) may help the supervisor or auditor to detect classification or other types of errors made by specific personnel or may reveal a pattern of overrides in entries that changes some present interpretations of accounting, e.g., a specific division. Indeed, most of the new audit tools defined below strengthen the evidence surrounding accounting internal control. The basic organization of identical entries changes some present interpretations of accounting. For instance:

Aggregation - The summaries posting a given account report the monetary amount which each summary contributes to the account total. Thus, the exact amount of transaction flow is known, so judgment is improved.

Transaction flow - 100% of the general journal is identified and organized. Each summary represents a transaction flow. This is new information to the Accounting industry.

Transaction flow materiality - Long, complex entries can aggregate to material amounts, even though the individual ledger postings may not.

Interim flow reports - Often one side of an account range will contain journal entries which, when recapitulated, provide useful flow reports. Examples include interim cash flow and cost flow reports.

Design and Construction of the Matrix Data Cube

Figure 1

Previous matrix accounting literature has considered financial statements and has used a 2-dimensional, usually square, depiction. Most journal entries have more than two accounts. Thus, a conceptual matrix of n dimensions is required, where n is the number of specific account-sides in the entry (duplicate postings to a single account-side are combined). Figure 1 illustrates specific classification of a taxable sale, such as:

Assets (debit) 530.00
Liabilities (credit) 30.00
Revenues (credit) 500.00

Each combination of account-sides defines a specific cell in the matrix. It follows that a greater number of dimensions relates perfectly with the number of account-sides addressed; because of this, a matrix is not needed in actual records. We recommend that accounts always appear in ascending number sequence, resulting in combinations only, permutations not allowed. The resulting record of combinations is both manageable and sortable. This classification scheme is a true "datacube", unlike computer usage of the same term where it applies to a rectangular solid.

Example of Journal Entry Traffic Report and Subsequent (Receipts) Cash Flow Report

Figure 2
Figure 2 lists 16 cash accounts of the publicly-available thinsft7 transaction set. A number of different users have placed entries on this set, resulting in many illogical amounts. Account 1010000, Cash in Bank/Main Office, (although low turnover) provides an adequate illustration. We will concentrate on the nine patterns of cash receipt.

Figure 3
Figure 3, result of the debit side single-account search, lists the nine summaries, with notations that #81 has the most transactions and that #85 is the only compound deposit. (Early field usage suggests that summary should also display the search-item [dr. Cash, 1010000 in this case] dollar amount.)

Figure 4
Figure 4 displays the only compound, with two transactions - which will hopefully appear in the same manner as written in the database/file for that transaction. We suggest that the database/file be converted to human readability before the index is built.

Figure 5
Figure 5 is the recapitulation of the nine debit summaries listed in Figure 3. In this simple case, only one elimination is required to organize sources for an interim cash flow statement.

Management of the Accounting Process

Range of account-side (b) The list created by the range search forms the basic data for a flow report.

Recapitulation This converts a string of journal entries to a single entry of totals of all the account-sides in the string. Useful for consolidating data for flow reports.

Supervision Individual transactions flows often begin at a specific point. Matrix selection of journal entries usually displays a group of related entries, allowing management to identify where errors are being made and to make corrections nearer to the source of the transaction.

Expected New Searches

near neighbor: Following a unit-of-freedom approach, a 20-line entry, possibly reporting a spreadsheet, would have three possible one-unit-of-freedom neighbors:

mirror-image: The exact opposite of a specific entry. This might be a cancellation of an earlier entry or an elimination in the consolidation process.


AICPA. 2003. Practice Alert 2003-2.

AICPA. 2002. Statement of Audit Standards 99. CPA2Biz,inc: Jersey City, NJ. DeMorgan, A. 1846. Elements of Arithmetic (5th ed.). Taylor and Walton: London.

Leech, S. A. 1986. The Theory and Development of a Matrix-Based Accounting System. Accounting and Business Research, (Autumn).

Mattesich, R. 1964. Accounting and Analytical Methods. Richard D. Irwin: Scarborough, ON.

Salton, G,; A. Wong, & C. S. Yang. 1975 A Vector Space Model for Automatic Indexing. Communications of the ACM, 18 (11): 613 - 620.

Sampson, W. C. 1995. Method and Electronic Apparatus for Performing Bookkeeping. U.S. Patent No. 5,212,639.

Sampson, W. C. 1996. Transaction Index: A Tool for Auditors. Internal Auditing, (Spring): 16 - 24.

Sampson, W. C. and M. J. Olan. 1993. Method and Electronic Apparatus for the Classification of Combinatorial Data for the Summarization and/or Tabulation Thereof. U.S. Patent No. 5,212,639.

Sampson, W. C. and D. Z. Douthat. 2000. Method for the Organizational Indexing, Storage, and Retrieval of Data According to Data Pattern Signatures. U. S. Patent No. 6,058,392.

Sorter, G. H. 1969. An Events Approach to Basic Accounting Theory. The Accounting Review, 44 (1):12-19.