Operational matrices are fundamental tools in quantitative techniques for accountancy, enabling professionals to analyse and represent complex operational data systematically. Mastery of these matrices facilitates efficient decision-making, cost control, and performance evaluation within accounting departments of Kenyan institutions such as banks, SACCOs, and county government offices. This chapter focuses on understanding the operational matrices through their key terms, definitions, and mathematical applications, preparing accountancy students to apply these concepts effectively in their professional roles.

4.1 Meaning of terms

Operational matrices involve structured arrangements of numerical data that represent operational relationships and quantities in a tabular form. Understanding the terminology associated with these matrices is essential for accurate interpretation and application in accountancy.

4.1.1 Matrix

A matrix is a rectangular array of numbers arranged in rows and columns, used to represent data or coefficients in a structured form. The size of a matrix is described by the number of rows \(m\) and columns \(n\), denoted as an \(m \times n\) matrix. Matrices are used in accounting to organise financial data such as cost allocations, transaction records, and operational outputs.

The general form of a matrix \(A\) is:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

Worked Examples

Example 1: Given a matrix representing cost centers in a retail business with 2 departments and 3 types of expenses:

$$ A = \begin{bmatrix} 15000 & 12000 & 8000 \\ 10000 & 9000 & 7000 \end{bmatrix} $$

Identify the dimensions of matrix \(A\).

Given: Number of rows \(m = 2\), number of columns \(n = 3\)

Answer: Matrix \(A\) is a \(2 \times 3\) matrix.

Example 2: A SACCO records loan disbursement amounts over 3 months across 4 branches as:

$$ L = \begin{bmatrix} 500000 & 600000 & 550000 & 580000 \\ 520000 & 610000 & 560000 & 590000 \\ 530000 & 620000 & 570000 & 600000 \end{bmatrix} $$

What is the size of matrix \(L\)?

Given: Rows \(m = 3\), Columns \(n = 4\)

Answer: Matrix \(L\) is a \(3 \times 4\) matrix.

Example 3: Construct a \(2 \times 2\) matrix \(B\) with the following elements: \(b_{11} = 1000\), \(b_{12} = 1500\), \(b_{21} = 2000\), \(b_{22} = 2500\).

$$ B = \begin{bmatrix} 1000 & 1500 \\ 2000 & 2500 \end{bmatrix} $$

Answer: Matrix \(B\) is as shown above.

4.1.2 Square Matrix

A square matrix is a matrix with an equal number of rows and columns, denoted as \(n \times n\). In accounting, square matrices often represent relationships such as input-output tables where operations are balanced across entities.

The general form of a square matrix \(S\) is:

$$ S = \begin{bmatrix} s_{11} & s_{12} & \cdots & s_{1n} \\ s_{21} & s_{22} & \cdots & s_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ s_{n1} & s_{n2} & \cdots & s_{nn} \end{bmatrix} $$

Worked Examples

Example 1: Verify if the following matrix from a county government budget allocation is square:

$$ M = \begin{bmatrix} 300000 & 250000 \\ 400000 & 350000 \end{bmatrix} $$

Given: Rows \(m = 2\), Columns \(n = 2\)

Answer: Matrix \(M\) is a \(2 \times 2\) square matrix.

Example 2: A hotel records monthly expenses for 3 departments over 3 categories:

$$ H = \begin{bmatrix} 12000 & 15000 & 13000 \\ 14000 & 16000 & 13500 \\ 12500 & 15500 & 14000 \end{bmatrix} $$

Determine if \(H\) is square.

Given: \(m = 3\), \(n = 3\)

Answer: Matrix \(H\) is a \(3 \times 3\) square matrix.

Example 3: Given matrix

$$ N = \begin{bmatrix} 10000 & 12000 & 11000 \\ 9000 & 10000 & 9500 \end{bmatrix} $$

Is \(N\) square?

Given: \(m = 2\), \(n = 3\)

Answer: Matrix \(N\) is not square.

4.1.3 Zero Matrix

A zero matrix is a matrix in which all elements are zero. This matrix represents the absence of any operational quantity or relationship. In accountancy, it can represent a scenario where no transactions or costs are incurred.

The zero matrix of size \(m \times n\) is:

$$ O = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix} $$

Worked Examples

Example 1: Write the zero matrix of size \(2 \times 3\).

$$ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

Answer: The zero matrix \(O\) is as above.

Example 2: A bank's loan disbursement matrix shows no loans in a quarter for 3 branches and 4 loan types. Represent this as a zero matrix.

Given: Size \(3 \times 4\)

$$ O = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$

Answer: The zero matrix \(O\) is as above.

Example 3: Confirm if the following matrix is a zero matrix:

$$ P = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $$

Check elements.

Answer: \(P\) is not a zero matrix because \(p_{22} = 1 \neq 0\).

4.1.4 Identity Matrix

An identity matrix is a special square matrix with ones on the main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. This property is useful in accounting software when verifying data integrity or performing matrix-based calculations.

The identity matrix of order \(n\) is:

$$ I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} $$

Worked Examples

Example 1: Write the identity matrix of order 3.

$$ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Answer: The identity matrix \(I_3\) is as above.

Example 2: Multiply the following matrix by the identity matrix \(I_2\):

$$ A = \begin{bmatrix} 5000 & 3000 \\ 2000 & 1000 \end{bmatrix}, \quad I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$

Formula:

$$ A \times I_2 = A $$

Calculation:

$$ \begin{bmatrix} 5000 & 3000 \\ 2000 & 1000 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} (5000 \times 1 + 3000 \times 0) & (5000 \times 0 + 3000 \times 1) \\ (2000 \times 1 + 1000 \times 0) & (2000 \times 0 + 1000 \times 1) \end{bmatrix} $$

$$ = \begin{bmatrix} 5000 & 3000 \\ 2000 & 1000 \end{bmatrix} $$

Answer: The product is matrix \(A\) unchanged.

Example 3: Verify if the following matrix is an identity matrix:

$$ J = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} $$

Check diagonal and off-diagonal elements.

Answer: \(J\) is not an identity matrix because \(j_{23} = 1 \neq 0\).

Chapter Summary

This chapter introduced the fundamental terms related to matrices, clarifying their structure and components. It then explored various types of matrices, including square, rectangular, diagonal, and identity matrices. The concept of determinants was covered with a focus on calculating determinants for 2 by 2 and 3 by 3 matrices, which are essential for solving matrix equations. The process of finding the inverse of a matrix was explained, highlighting its importance in matrix operations and problem solving. Practical applications of matrices were discussed, demonstrating how they can be used to model and solve real-world problems. Finally, the chapter covered time series analysis, showing how operational matrices can be applied to analyze data trends over time. This comprehensive overview equips students with both theoretical knowledge and practical skills in working with matrices and their applications.