# Accounting Games: Using Matrix Algebra in Creating the Accounting Models

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Matrix Accounting Modeling Using Blocks of Matrices

- Record transactions and present the Ledger on their basis in the form of equivalent matrices.
- Transform the initial data into Trial Balances corresponding to their equivalents in the system of matrix algebra transactions.
- Relate the opening and closing account balances by means of the basic accounting equation in matrix form.
- Determine the formulae to receive the trial Balance.
- Determine a matrix model as a system of equivalents of the data presented by corresponding Trial Balances.

**Definition**

**1.**

**Definition**

**2.**

_{x}

_{,y}· E(X,Y).

**LM**), as follows:

_{i}is the sum of transactions corresponding to entry i, and

**E**(X

_{i},Y

_{i}) is a correspondence matrix referring to the entry i.

**MDT**) is obtained from the Ledger Matrix (

**LM**), as follows:

**MCT**) is a transposed matrix

**MDT**, as follows:

_{1}, c

_{2}, …, c

_{m}are accounting symbols used to represent the matrix model of an accounting system, where ${S}_{X,Y}={\displaystyle \sum _{{i}_{XY}=1}^{{n}_{X,Y}}{S}_{{i}_{XY}}}$ is the total sum of the transaction referred to the accounts X,Y. In this case, $\sum _{X={c}_{1}}^{{c}_{m}}}{\displaystyle \sum _{Y={c}_{1}}^{{c}_{m}}{n}_{XY}}=n$, where n is the total number of entries in the Ledger.

- AA—the matrix of “assets-assets” transactions;
- AL—the matrix of “assets-liabilities” transactions;
- LA—the matrix of “liabilities- assets” transactions;
- LL—the matrix of “liabilities-liabilities” transactions;
- CA—the matrix of “capital-assets” transactions;
- CC—the matrix of “capital-capital” transactions.

**MTB**) using the ALC-grouping is represented as follows. (Here the “0” subscript sign means the beginning of the period t − 1 = 0, the “1” sign means the end of the period t = 1.):

**MCT**) contains the data from the transactions with inverted correspondences of the accounts used.

## 4. Accounting Game by Means of Matrix Modeling

- January, 5—player I wins $20 from the player II.
- Player III wins $8 from player II on the same date.
- January, 8—player II wins $63 from player I, etc.

## 5. The Solution

∆S (I, II) = S (I,II) – S (II,I) = 38 − 6 = +32 > 0,

∆S (II,I) = S (II,I) – S (I,II) = 6 − 38 = −32 < 0

**MDT**). Thus, the chess balance resembles the liability matrix (to receive) and can be represented as follows (Table 4):

**MCT**) (Table 5).

**MCT**) from a “liabilities to receive” matrix, or Matrix of Debit Turnovers (

**MDT**) in order to obtain a Trial Balance Matrix (

**MTB**) as follows:

**MDT**−

**MCT**=

**MTB**

**MTB**can be represented in matrix form as follows (Table 6):

**MDT**is a matrix of debit turnovers.

**MCT**is a transposition of

**MDT (**$MD{T}^{\text{'}}$

**),**or a matrix of credit turnovers.$MT{B}_{1}$ is a Trial Balance matrix for the end of the period.

## 6. Conclusions

## Funding

## Conflicts of Interest

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# | Date | Liabilities to Receive | Liabilities to Pay | Total ($) |
---|---|---|---|---|

1 | 5.01 | I | II | 20 |

2 | 5.01 | III | I | 8 |

3 | 8.01 | II | I | 6 |

4 | 16.01 | I | III | 14 |

5 | 18.01 | II | III | 16 |

6 | 25.01 | III | II | 12 |

7 | 29.01 | I | II | 18 |

8 | 31.01 | III | I | 6 |

Total | 100 |

Liabilities | Sum (c.u.) | |
---|---|---|

To Receive | To Pay | |

I | II | 20 |

# | Liabilities to Receive | Liabilities to Pay | Total ($) |
---|---|---|---|

1 | I | II | 38 |

2 | III | I | 14 |

3 | II | I | 6 |

4 | I | III | 14 |

5 | II | III | 16 |

6 | III | II | 12 |

Total | 100 |

Type of Matrix | To Receive | To Pay | Total to Receive | ||
---|---|---|---|---|---|

I | II | III | |||

MDT= | I | 0 | 38 | 14 | 52 |

II | 6 | 0 | 16 | 22 | |

III | 14 | 12 | 0 | 26 | |

Total to pay | 20 | 50 | 30 | 100 |

Type of Matrix | To Receive | To Pay | Total to Receive | ||
---|---|---|---|---|---|

I | II | III | |||

MCT= | I | 0 | 6 | 14 | 20 |

II | 38 | 0 | 12 | 50 | |

III | 14 | 16 | 0 | 30 | |

Total to pay | 52 | 22 | 26 | 100 |

Type of Matrix | To Receive | To Pay | Total to Receive | ||
---|---|---|---|---|---|

I | II | III | |||

MTB= | I | 0 | +32 | 0 | +32 |

II | −32 | 0 | +4 | −28 | |

III | 0 | −4 | 0 | −4 | |

Total to pay | −32 | +28 | +4 | 0 |

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Vysotskaya, A. Accounting Games: Using Matrix Algebra in Creating the Accounting Models. *Mathematics* **2018**, *6*, 152.
https://doi.org/10.3390/math6090152

**AMA Style**

Vysotskaya A. Accounting Games: Using Matrix Algebra in Creating the Accounting Models. *Mathematics*. 2018; 6(9):152.
https://doi.org/10.3390/math6090152

**Chicago/Turabian Style**

Vysotskaya, Anna. 2018. "Accounting Games: Using Matrix Algebra in Creating the Accounting Models" *Mathematics* 6, no. 9: 152.
https://doi.org/10.3390/math6090152