MJ-8 | Business Mathematics

and Business Statistics:

Part I: Business Mathematics
Chapter 1: Matrix Algebra and Determinants
Matrices are a foundational tool in business and economics, providing an efficient
framework for modelling complex systems with multiple interacting variables. A
matrix is formally defined as a rectangular array of numbers or other
mathematical objects arranged in rows and columns.1 Its dimensions are
described by the number of rows (m) and the number of columns (n), denoted as
an m x n matrix. The rows are the horizontal entries, while the columns are the
vertical ones.1 This structured representation is invaluable for organizing and
manipulating large datasets, solving systems of linear equations, and optimizing
business processes.3

1.1 Definition and Taxonomy of Matrices

The classification of matrices is based on their structure and properties, which
dictate their application in various contexts.
• Row and Column Matrices: A matrix with a single row is known as a row
matrix or row vector, while a matrix with a single column is called a column
matrix or column vector.2 These are often used to represent vectors or
single-dimensional data sets.
• Square Matrix: A square matrix has an equal number of rows and
columns.2 This type of matrix is particularly important because it is a
prerequisite for many advanced operations, such as finding a determinant
or an inverse.
• Identity Matrix: An identity matrix, denoted by I, is a special type of
square matrix with ones on its main diagonal (from the top-left to the
bottom-right) and zeros everywhere else.2 It functions as the multiplicative
identity in matrix algebra, similar to how the number 1 functions in scalar
arithmetic. For any matrix
A, the relationship AI=IA=A holds true.5
• Invertible Matrix: An invertible or non-singular matrix is a square matrix
for which an inverse matrix exists.2 The existence of an inverse is
contingent on the matrix having a non-zero determinant.
• Other Types: Other specific types of matrices include diagonal matrices
(where all non-diagonal elements are zero) and triangular matrices (where
all elements above or below the main diagonal are zero).2
The applications of matrices are wide-ranging. They are used in financial analysis
for portfolio optimization and risk assessment, in supply chain management for
modelling networks and resource allocation, and in operations research to find
optimal solutions that adhere to various constraints.4

1.2 Fundamental Matrix Operations: Addition, Subtraction, and

Multiplication
Matrix algebra is governed by a specific set of rules for performing operations.
• Addition and Subtraction: These operations can only be performed on
matrices of the same size. The addition or subtraction is carried out by
adding or subtracting the corresponding elements of the two matrices.8
• Multiplication: Matrix multiplication is a more complex operation that
requires a specific condition to be met, known as the conformability rule.
For the product of two matrices, A and B, to exist, the number of columns in
the first matrix (A) must be equal to the number of rows in the second
matrix (B). If A is an m×n matrix and B is an n×p matrix, the resulting
product, C=AB, will be an m×p matrix.5 The element in the ith row and jth
column of C is found by multiplying the elements of the ith row of A with
the corresponding elements of the jth column of B and summing the
products.5
A common application of this rule in business is the calculation of total sales or
revenue, where a price matrix is multiplied by a quantity matrix.8

1.3 Determinants: Properties and Calculation

The determinant is a scalar value that is a function of the entries of a square
matrix.2 It is denoted by ∣A∣ or det(A). A key property of the determinant is its
role in determining the invertibility of a matrix: a square matrix has an inverse if
and only if its determinant is non-zero.2
• For a 2x2 matrix: A=[acbd], the determinant is calculated as ∣A∣=ad−bc.10

• For a 3x3 matrix: A= adgbehcfi , the determinant is calculated

using the cofactor expansion method along a row or column, such as the
first row: ∣A∣=a(ei−fh)−b(di−fg)+c(dh−eg).10

1.4 Matrix Inversion: Methods and Conditions

The inverse of a square matrix A, denoted by A−1, is a matrix that, when
multiplied by A, yields the identity matrix: A⋅A−1=A−1⋅A=I.6

The two conditions for a matrix inverse to exist are:

1. The matrix must be square.
2. Its determinant must be non-zero.6
The most common method for finding the inverse of a matrix is the adjoint
method.6 This involves a series of steps:
 1. Calculate the minors for all elements of the matrix. A minor of an element is
the determinant of the smaller matrix obtained by deleting the row and
column containing that element.6
2. Compute the cofactor of each element. The cofactor is the minor multiplied
by (−1)i+j, where i and j are the row and column indices of the element.7
3. Form the adjoint matrix, which is the transpose of the cofactor matrix.6
4. Finally, divide the adjoint matrix by the determinant of the original matrix:
A−1=∣A∣1⋅Adj(A).6
Consider a 2x2 matrix, A=[7439], used in a business problem to solve for
equilibrium prices.8
• The determinant is ∣A∣=(7)(9)−(3)(4)=63−12=51.8
• The adjoint of a 2x2 matrix is found by swapping the main diagonal
elements and negating the off-diagonal elements: Adj(A)=[9−4−37].8
• The inverse is A−1=511[9−4−37].8

1.5 Applications of Matrices in Business and Economics

Matrices are not just theoretical constructs; they are practical tools used to model
and solve real-world business problems.3
• Solving Systems of Linear Equations: Matrices provide a compact and
powerful way to represent and solve systems of linear equations, which
often arise in business contexts. For example, a system of equations can
model the equilibrium prices in a market with interdependent goods, and
the matrix inversion method can be used to find the solution.8
• Resource Allocation: Businesses use matrices to model the allocation of
limited resources across multiple production lines or projects to maximize
overall efficiency, often with the help of linear programming techniques.4
• Portfolio Optimization: In finance, matrix algebra is used to model
portfolio risk and returns. The covariance matrix of asset returns is used to
formulate an optimization problem, which is then solved using matrix
inversion to determine the optimal asset weights that balance risk and
return.4
• Leontief Input-Output Model: This model uses matrices to analyze the
interdependencies between different sectors of an economy. It helps
determine the production levels required from each sector to meet final
demand.12
 Chapter 2: Introduction to Differential Calculus
Differential calculus is a branch of mathematics concerned with rates of change
and slopes of curves. The study of functions, limits, and continuity provides the
essential foundation for understanding and applying differentiation in business.

2.1 Mathematical Functions

A function describes a relationship between an input (independent variable) and
a single output (dependent variable).13 In business, functions are used to model
relationships between different variables, such as cost, revenue, and profit.
Common types include linear, quadratic, and polynomial functions.15 For
instance, a simple linear function might model total cost as a function of
production volume, while a more complex polynomial function could represent
total revenue.

2.2 Limits of a Function

A limit describes the value that a function approaches as its input variable gets
arbitrarily close to a specific value.13 This concept is crucial for defining
continuity and differentiation. The notation for a limit is
limx→af(x)=L, which states that the value of the function f(x) approaches L as x
approaches a.13

2.3 Continuity
A function is considered continuous if its graph can be drawn without lifting the
pen from the paper, meaning it has no breaks, holes, or jumps.13 For a function to
be continuous at a specific point, x=a, three conditions must be met:
1. f(a) must be defined.13
2. The limit of the function as x approaches a must exist, i.e., limx→af(x)
exists.13
3. The value of the function at the point must equal its limit: limx→af(x)=f(a).13
The concepts of limits and continuity are more than just theoretical exercises;
they are preconditions for differentiability.14 A function must be continuous to be
differentiable, and differentiability allows for the calculation of the instantaneous
rate of change (the derivative).14 In a business context, this is the basis for
marginal analysis—for example, calculating the marginal cost or marginal
revenue. It also enables optimization problems, such as finding the production
level that maximizes profit, by locating the maximum point on the profit function
curve.15
 Chapter 3: Foundational Mathematics of Finance
This section explores the core financial concepts of interest rates and valuation,
which are essential for making informed business and investment decisions.

3.1 Simple and Compound Interest

Interest is the cost of borrowing money or the return on an investment.16
• Simple Interest: Simple interest is calculated solely on the original
principal amount of a loan or investment.16 It is a straightforward
calculation that does not account for accumulated interest. The formula for
simple interest is: Simple Interest
=P×i×n, where P is the principal, i is the interest rate, and n is the term.16
• Compound Interest: Compound interest is calculated on both the initial
principal and all of the interest that has accumulated from previous
periods, often referred to as "interest on interest".16 The compounding
effect can lead to a significantly higher return or cost over time compared
to simple interest.16 The formula for the future value (FV) with compound
interest is: FV=PV×(1+ni)(n×t), where PV is the present value, i is the
interest rate, n is the number of compounding periods per year, and t is the
total time.16

3.2 Interest Rates: Nominal and Effective

• Nominal Rate: The nominal interest rate is the stated or advertised rate on
a loan or investment, without considering the effect of compounding.12
• Effective Rate: The effective interest rate is the actual rate of return or cost
after taking into account the effects of compounding over a given period.12
Because compounding applies interest to the accrued interest, the
effective rate is generally higher than the nominal rate when interest is
compounded more than once a year. The formula for the effective interest
rate, i.e., is related to the nominal rate, in, by the number of compounding
periods, m, per year: i.e.,=(1+min)m−1.12

3.3 Compounding and Discounting

• Compounding: Compounding is the process of calculating the future
value of a current sum of money, assuming a certain interest rate.16 It
allows for the forecasting of growth in savings or debt.
• Discounting: Discounting is the inverse of compounding. It is the process
of finding the present value of a future sum of money.16 This is a crucial
concept for valuing assets, as it allows for the determination of what a
future cash flow is worth today, given a specified rate of return. The
formula for present value (PV) is: PV=(1+r)tFV.16
 Part II: Business Statistics
Chapter 4: Univariate Analysis: Describing Data
Univariate analysis focuses on describing a single variable within a data set.
Measures of central tendency and dispersion are the two primary tools for this
type of analysis.

4.1 Measures of Central Tendency: Mean, Median, and Mode

Measures of central tendency provide a single, representative value for a data
set. The most common measures are the mean, median, and mode.18
• Mean: The arithmetic mean, or average, is the sum of all data points
divided by the total number of observations.19 For grouped data, the mean
(xˉ) is calculated as xˉ=∑fi∑fixi, where xi is the midpoint of each class and
fi is the frequency.19
• Median: The median is the middle value in a data set that has been
arranged in ascending or descending order.18 The median is a positional
average, and it is also known as the second quartile (Q2).21
• Mode: The mode is the value that appears most frequently in a data set.18

Solved Example: Measures of Central Tendency

We can calculate the mean and median for the grouped data:

Frequency Midpoint Cumulative Frequency

Class fx
(f) (x) (cf)

0-5 2 2.5 5 2

5-10 9 7.5 67.5 11

10-15 15 12.5 187.5 26

15-20 10 17.5 175 36

20-25 5 22.5 112.5 41

25-30 9 27.5 247.5 50

Total ∑f=50 ∑fx=795

Mean: xˉ=∑f∑fx=50795=15.9

Median:
• First, find the median class. The median is the value of the 2Nth term, which
is 250=25.19 The 25th value falls in the 10-15 class, as its cumulative
frequency (26) is the first to exceed 25.
• Use the formula for grouped data: Median=l+f2N−cf×h.19
o l = lower limit of the median class = 10
o N = total frequency = 50
o cf = cumulative frequency of the class preceding the median class =
11
o f = frequency of the median class = 15
o h = class interval = 5

• Median=10+1525−11×5=10+1514×5=10+4.67=14.67

4.2 Measures of Dispersion: Quantifying Data Spread

Measures of dispersion quantify the spread or variability of a data set.20
• Quartile Deviation (QD): Also known as the semi-interquartile range, QD
measures the spread of the middle 50% of the data.21 It is calculated as half
the difference between the third quartile (Q3) and the first quartile (Q1):
Q.D.=2Q3−Q1.21 The quartile values are found using specific formulas for
grouped and ungrouped data.21
• Standard Deviation (SD): The standard deviation is a widely used
measure that quantifies the amount of variation or dispersion of a set of
data values from the mean.20 It is the square root of the variance.
For a population, the formula is σ=N∑(x−μ)2, and for a sample, it is
s=n−1∑(x−xˉ)2.24

For a frequency distribution, the formula is σ=∑f∑f(x−xˉ)2.25

The squaring of deviations gives more weight to outliers than other

measures like mean deviation.20
The exam image contains a true/false statement: "Standard deviation is always
less than mean deviation" [Image 1]. This statement is generally false. Standard
deviation, by its formula, squares the deviations from the mean before summing
them, while mean deviation uses absolute values.22 This mathematical operation
gives disproportionate weight to larger deviations, causing the standard
deviation to be equal to or greater than the mean deviation in most distributions.
 Chapter 5: Bi-variate Analysis: Exploring
Relationships
Bivariate analysis examines the relationship between two variables. Correlation
and regression analysis are the main techniques for this.

5.1 Introduction to Correlation and Regression

• Correlation: Correlation measures the degree to which two variables are
related. It quantifies the strength and direction of a linear relationship.15
• Regression: Regression analysis goes a step further by using the
relationship between two variables to predict the value of one (dependent
variable) from the value of the other (independent variable).15

5.2 Simple and Linear Correlation: Karl Pearson's Coefficient

Karl Pearson's coefficient of correlation (r) is a numerical measure of the strength
of a linear relationship between two variables.28 The value of
r ranges from -1 to +1. A value of +1 indicates a perfect positive linear
correlation, -1 indicates a perfect negative linear correlation, and 0 indicates no
linear correlation.28
The formula is given by: rx,y=[n(∑x2)−(∑x)2][n(∑y2)−(∑y)2]n(∑xy)−(∑x)(∑y).29

5.4 Other Methods of Correlation

• Spearman's Rank Correlation: This method is used to measure the
correlation between two sets of ranked data. It is a non-parametric
measure, useful when the data is ordinal or when a linear relationship
cannot be assumed.15
• Concurrent Deviation Method: This is a simple, non-rigorous method that
focuses only on the direction of change in the variables.32 The coefficient is
based on the number of times the deviations of two variables from their
previous values have the same sign.
• Fisher's Z-Transformation: This is a transformation of Pearson's
correlation coefficient, not a method for calculating correlation itself.33 The
transformation converts the sampling distribution of Pearson's
r into a variable that is approximately normally distributed, which is useful
for statistical inference, such as creating confidence intervals and testing
hypotheses about the population correlation coefficient.33
Fisher's coefficient is a post-calculation transformation rather than a primary
method for deriving the correlation coefficient from raw data.
 Chapter 6: Index Numbers
Index numbers are statistical tools used to measure relative changes in a variable
or a group of related variables over time or across different locations.35 They are
expressed as a percentage, with one period serving as a base for comparison.35
6.1 Meaning and Uses of Index Numbers
Index numbers are used to compare levels of a phenomenon where direct
measurement is not feasible. For example, the general price level in an economy
is an abstract concept that can be measured using a price index number.36 They
are widely used in economics and business to:
• Measure changes in the general price level (e.g., Consumer Price Index,
Wholesale Price Index).35
• Track changes in production, foreign trade, and stock market
performance.35
• Inform economic policy and business decisions by summarizing complex
changes into a single, understandable value.35

6.2 "Index numbers are economic barometers"

Index numbers measure changes in key economic variables to gauge the health
and direction of the economy.35 For example:
• A rising Consumer Price Index (CPI) acts as a barometer for inflation,
signaling that the cost of living is increasing. This information is critical for
central banks, which may use it to decide whether to raise interest rates to
cool down the economy.36
• A positive trend in an Industrial Production Index can be a barometer for
economic growth, indicating that manufacturing and other industrial
sectors are expanding.35
In essence, index numbers serve as diagnostic and predictive tools for
economists and policymakers, providing them with a concise summary of
complex economic conditions.
 Chapter 7: Time Series Analysis
Time series analysis is a statistical method for analyzing data points collected
over successive time intervals to identify patterns and make predictions.37
7.1 Introduction to Time Series
A time series is a sequence of data points indexed in time order.37 It is an
indispensable tool in finance, economics, and operations for forecasting future
trends based on historical data.37

7.2 Components of a Time Series

A time series is composed of several components that contribute to its overall
pattern.37 These are often represented by additive (Yt=Tt+St+Ct+Rt)
or multiplicative (Yt=Tt×St×Ct×Rt) models, where Yt is the observed value at
time t.38
• Trend (Tt): This represents the long-term, underlying movement in the
data. A trend can be upward, downward, or remain constant over time.37
For example, a consistent annual increase in bicycle sales over a decade
would indicate a positive trend.38
• Seasonality (St): This refers to periodic variations that repeat at fixed
intervals, such as a year, month, or week. These patterns are often
influenced by predictable factors like seasons or holidays. For instance,
bicycle sales often peak during the summer due to favorable weather
conditions.38
• Cyclical (Ct): These are fluctuations that are typically longer in duration
than seasonal patterns and are irregular and difficult to predict.38
Economic cycles, such as recessions or booms, are common drivers of
cyclical variations in business data.38
• Irregular/Random (Rt): This component represents the unpredictable
and random movements in a time series that are not explained by the
trend, seasonality, or cyclical components.37

7.3 Trend Analysis

Trend analysis involves estimating and modeling the long-term movement in a
time series. Two common methods are:
• Moving Average Method: This method smooths out short-term
fluctuations and irregular variations by calculating the average of a fixed
number of data points.38 The result is a smoother line that reveals the
underlying trend.
• Least Squares Method: This is a statistical method used to fit a straight
line (or other curve) to a set of data points, minimizing the sum of the
squared differences between the data points and the line.39 The resulting
linear trend line provides a simple model of the long-term direction of the
data.
 Part III: Supplemental Topics and
Problem-Solving
.
Chapter 8: Progressions (Arithmetic and Geometric)
A progression is a sequence of numbers arranged in a specific order based on a
rule.

8.1 Arithmetic Progressions (AP)

An arithmetic progression is a sequence where the difference between
consecutive terms is constant. This constant difference is known as the common
difference (d).40
• nth Term Formula: The formula for the nth term (an) is given by an
=a+(n−1)d, where a is the first term.40
• Sum of n Terms Formula: The sum of the first n terms (Sn) is given by Sn
=2n[2a+(n−1)d] or Sn=2n(a+l), where l is the last term.40
Solved Example: AP
Problem 1: Find the 12th term of the arithmetic progression 3, 5, 7,...
• The first term is a=3.
• The common difference is d=5−3=2.
• Using the formula an=a+(n−1)d, the 12th term is
a12=3+(12−1)2=3+11(2)=3+22=25.40
Problem 2: The sum of 3 numbers in AP is 15 and the product of the first and the
last is 21. Find the numbers.
• Let the three numbers be a−d,a,a+d.
• Sum: (a−d)+a+(a+d)=3a=15⇒a=5.
• Product of first and last: (a−d)(a+d)=a2−d2=21.
• Substituting a=5: 52−d2=21⇒25−d2=21⇒d2=4⇒d=±2.
• If d=2, the numbers are 5−2,5,5+2⇒3,5,7.
• If d=−2, the numbers are 5−(−2),5,5+(−2)⇒7,5,3.

8.2 Geometric Progressions (GP)

A geometric progression is a sequence where each term is found by multiplying
the previous term by a constant value. This constant value is known as the
common ratio (r).40
• nth Term Formula: The formula for the nth term (an) is given by an
=arn−1, where a is the first term.40
 • Sum of n Terms Formula: The sum of the first n terms (Sn) is given by Sn
=r−1a(rn−1) for r>1 or Sn=1−ra(1−rn) for r<1.40
Solved Example: GP

Problem: Find the 5th term of the G.P. 3, 6, 12,....

• The first term is a=3.

• The common ratio is r=6/3=2.
• Using the formula an=arn−1, the 5th term is a5=3(2)5−1=3(2)4=3(16)=48.40
 Chapter 9: Data Collection Methods
Data collection is the process of gathering and measuring information from
various sources to answer research questions.46 There are two primary types of
data: primary and secondary.

9.1 Primary vs. Secondary Data

• Primary Data: This is raw, firsthand information collected directly from
the original source.46 It is unique to the specific research project and has
not been influenced or altered by others.46 Primary data is essential for
gaining specific, tailored insights for a unique business problem, such as
conducting a market survey for a new product.46
• Secondary Data: This is existing information that has already been
collected and published by someone else for a different purpose.46 It is not
original to the current research but can be used to support an argument,
provide background context, or inform strategic decisions.48
Distinction between Primary and Secondary Data:
The primary difference lies in the nature of the information itself. Primary data is
raw and provides direct access to the subject of the research, while secondary
data is second-hand and provides commentary or interpretation from other
researchers. Primary data collection is typically more expensive and time-
consuming, while secondary data is readily available and less costly.

Methods of Collection
• Primary Methods: The most common methods include surveys,
interviews, focus groups, and experiments.46 Each method is designed to
gather specific information directly from a target audience or
phenomenon.46
• Secondary Methods: Secondary data is obtained from existing sources
such as the internet, government and non-government agencies, public
libraries, textbooks, and academic journals.50
 Appendix: Summary of
Formulas and Key Concepts