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Compiled by: Bonface Omondi
Statistical matrices are essential tools in business mathematics and statistics, widely used across Kenyan industries including banks, hospitals, retail businesses, and county government offices. They provide structured numerical summaries that help professionals analyze data, identify patterns, and make informed decisions. Understanding how to apply and interpret these matrices enables students and professionals to manage data effectively in diverse sectors such as healthcare, education, agriculture, and finance.
Statistical matrices organize data in a tabular form to reveal relationships among variables and facilitate numerical analysis. The order of a matrix refers to its dimensions, expressed as the number of rows by the number of columns, which determines how data is structured for analysis. Different types of matrices serve various purposes, such as representing raw data, summarizing frequencies, or showing correlations between variables. Mastery of these concepts is crucial for analyzing business data from organizations like SACCOs, retail chains, farms, and universities.
A matrix’s order is defined by the number of rows and columns it contains, written as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. The order dictates the matrix’s size and the type of data it can represent.
$$ \text{Order of matrix} = m \times n $$
Example 1: A county government office records monthly expenses for 4 departments over 3 months. What is the order of the matrix representing this data?
Given: 4 departments (rows), 3 months (columns)
$$ m = 4, \quad n = 3 $$
$$ \text{Order} = 4 \times 3 $$
Answer: 4 by 3 matrix
Example 2: A retail business tracks sales of 5 product categories across 7 stores. Find the order of the sales matrix.
Given: 5 categories (rows), 7 stores (columns)
$$ m = 5, \quad n = 7 $$
$$ \text{Order} = 5 \times 7 $$
Answer: 5 by 7 matrix
Example 3: A hospital collects data on 6 types of medical tests performed for 10 patients. Determine the matrix order.
Given: 6 tests (rows), 10 patients (columns)
$$ m = 6, \quad n = 10 $$
$$ \text{Order} = 6 \times 10 $$
Answer: 6 by 10 matrix
Matrices can be classified based on their shapes: row matrices have only one row, while column matrices have only one column. These types are useful for representing individual data vectors, such as sales figures or monthly attendance.
$$ \text{Row matrix order} = 1 \times n $$
$$ \text{Column matrix order} = m \times 1 $$
Example 1: A SACCO reports quarterly profits for one year. Represent this as a row matrix order.
Given: Profits for 4 quarters
$$ m = 1, \quad n = 4 $$
$$ \text{Order} = 1 \times 4 $$
Answer: Row matrix of order 1 by 4
Example 2: A university records the number of students enrolled in 8 courses. Represent this data as a column matrix.
Given: 8 courses enrollment numbers
$$ m = 8, \quad n = 1 $$
$$ \text{Order} = 8 \times 1 $$
Answer: Column matrix of order 8 by 1
Example 3: A farm tracks daily milk production for 1 cow over 30 days. Express as a row matrix order.
Given: 30 days data
$$ m = 1, \quad n = 30 $$
$$ \text{Order} = 1 \times 30 $$
Answer: Row matrix of order 1 by 30
A square matrix has equal numbers of rows and columns (\(m = n\)) and is often used in correlation and transition matrices. A rectangular matrix has unequal rows and columns (\(m \neq n\)) and typically represents data sets with different dimensions.
$$ \text{Square matrix order} = n \times n $$
$$ \text{Rectangular matrix order} = m \times n, \quad m \neq n $$
Example 1: A bank's credit risk matrix has 5 risk categories evaluated over 5 quarters. Identify the matrix type and order.
Given: 5 categories, 5 quarters
$$ m = 5, \quad n = 5 $$
$$ \text{Order} = 5 \times 5 $$
Answer: Square matrix of order 5 by 5
Example 2: A hotel records room occupancy for 10 room types over 12 months. Determine the matrix type and order.
Given: 10 types, 12 months
$$ m = 10, \quad n = 12 $$
$$ m \neq n $$
Answer: Rectangular matrix of order 10 by 12
Example 3: A cooperative tracks membership contributions for 7 branches across 7 years. What matrix type and order?
Given: 7 branches, 7 years
$$ m = 7, \quad n = 7 $$
$$ \text{Order} = 7 \times 7 $$
Answer: Square matrix of order 7 by 7
Special matrices have unique properties useful in computations:
Example 1: A NEMA report shows no environmental incidents in 4 regions over 4 months. Represent as a zero matrix.
Given: 4 regions, 4 months, no incidents
Zero matrix order:
$$ 4 \times 4 $$
All elements zero.
Answer: 4 by 4 zero matrix
Example 2: A hospital has 6 departments with budget allocations only on the main diagonal. Represent as a diagonal matrix.
Given: 6 departments
Diagonal matrix order:
$$ 6 \times 6 $$
Only diagonal entries non-zero.
Answer: 6 by 6 diagonal matrix
Example 3: A retail business uses a 5 by 5 identity matrix in inventory algorithm calculations. Write the matrix order and properties.
Given: 5 products
Identity matrix order:
$$ 5 \times 5 $$
Diagonal elements = 1, off-diagonal = 0
Answer: 5 by 5 identity matrix
A county referral hospital records patient visits by 8 departments over 6 months. Find the order of the matrix representing the data. (2 marks)
A SACCO tracks the monthly deposits of 12 members for 1 year. Represent the data as a matrix and state its order. (3 marks)
A retail store has 10 product categories and records weekly sales for 10 weeks. Identify the matrix type and order. (3 marks)
A university uses a zero matrix to represent no student attendance in 5 lecture halls over 5 days. Write the matrix order and describe its properties. (4 marks)
A farm uses a diagonal matrix to represent fertilizer application rates for 7 fields, with zero application elsewhere. What is the order of this matrix? (3 marks)
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Create a free accountThis chapter introduced matrices by explaining their order and the different types commonly used in business mathematics. It then covered the basic operations of addition, subtraction, and multiplication of matrices, emphasizing the rules that govern these processes. The concept of determinants was explored with a focus on 2x2 matrices, showing how to calculate their values. Following this, the chapter explained how to find the inverse of a 2x2 matrix, an essential tool for solving matrix equations. Finally, the practical application of matrices in business problems was demonstrated, illustrating how these mathematical tools can simplify and solve real-world business scenarios efficiently. Through these topics, the chapter provided a comprehensive foundation for understanding and applying statistical matrices in business contexts.
Calculate the sum of matrices
\[
A = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 7 \\ 6 & 3 \end{bmatrix}
\]
(2 marks)
Subtract matrix \( B \) from matrix \( A \), where
\[
A = \begin{bmatrix} 8 & 4 \\ 5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 2 \\ 1 & 7 \end{bmatrix}
\]
(2 marks)
Given matrices \(A = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 \\ 7 & 2 \end{bmatrix}\) representing monthly sales figures (in Ksh thousands) for two products at a Nairobi retail shop, find the matrix \(A + B\). (4 marks)
A county hospital records patient counts for two departments as matrices \(C = \begin{bmatrix} 30 & 45 \\ 25 & 35 \end{bmatrix}\) and \(D = \begin{bmatrix} 15 & 25 \\ 10 & 20 \end{bmatrix}\). Calculate the matrix \(C - D\). (4 marks)
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