By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world problems and make smart decisions in your trade and business.
Compiled by: Grace Nganga
Matrices are fundamental tools in various professional fields across Kenya, including finance, healthcare, agriculture, and administration. They provide an efficient way to organize and manipulate data, enabling informed decision-making and problem-solving. Mastery of matrix concepts and operations empowers professionals to analyze complex data sets, optimize resource allocation, and model real-world situations succinctly.
Matrices are rectangular arrays of numbers arranged in rows and columns, used to represent data or linear equations compactly. Each element in a matrix is identified by its position, denoted by row and column indices. Matrices are widely used in sectors such as banks for transaction data, hospitals for patient statistics, and county offices for demographic information.
A matrix is denoted by a capital letter such as \( A \), and its elements by lowercase letters with two subscripts indicating the row and column positions, for example, \( a_{ij} \) represents the element in the \( i^{th} \) row and \( j^{th} \) column.
$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$
where \( m \) is the number of rows and \( n \) is the number of columns.
Example 1: A retail business records monthly sales (in thousands of Ksh) for 2 products over 3 months as matrix \( S \):
Given:
$$ S = \begin{bmatrix} 15 & 20 & 25 \\ 10 & 18 & 22 \end{bmatrix} $$
Identify the element in the second row, third column.
Answer: \( a_{23} = 22 \) (Sales of product 2 in month 3 is 22,000 Ksh)
Example 2: Define matrix \( B \) with 3 rows and 2 columns:
$$ B = \begin{bmatrix} 5 & 8 \\ 7 & 3 \\ 6 & 4 \end{bmatrix} $$
State the order of matrix \( B \).
Answer: Matrix \( B \) is of order \( 3 \times 2 \) (3 rows, 2 columns).
Example 3: Given matrix \( C \) as
$$ C = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} $$
What is the element \( c_{12} \)?
Answer: \( c_{12} = 4 \)
Matrices are classified by their dimensions and element patterns. Common types include:
Example 1: Identify the type of matrix \( D \):
$$ D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix} $$
Answer: \( D \) is a diagonal matrix.
Example 2: Given
$$ E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
Identify the matrix type.
Answer: \( E \) is an identity matrix.
Example 3: Matrix \( F \) is:
$$ F = \begin{bmatrix} 4 & 7 & 9 \end{bmatrix} $$
Determine the order and type of \( F \).
Answer: \( F \) is a row matrix of order \( 1 \times 3 \).
The order of a matrix is its dimension expressed as rows × columns. The number of elements is the product of rows and columns. Elements are referred to by their position.
Example 1: Matrix \( G \) has 4 rows and 5 columns. Find the number of elements.
Answer:
$$ \text{Number of elements} = 4 \times 5 = 20 $$
Example 2: Find the order and number of elements of matrix \( H \):
$$ H = \begin{bmatrix} 2 & 5 \\ 7 & 8 \\ 1 & 3 \end{bmatrix} $$
Answer:
Order is \( 3 \times 2 \).
Number of elements:
$$ 3 \times 2 = 6 $$
Example 3: In matrix \( I \) of order \( 2 \times 4 \), what is the position of element 7 if it is at second row and third column?
Answer: Position is \( i = 2, j = 3 \), element \( i_{23} = 7 \).
Matrices are used to manage data such as employee records, financial transactions, production schedules, or patient statistics. For instance, a county referral hospital may use matrices to track patient admissions by department and month; a SACCO can record loan disbursements by member groups and months; an agricultural cooperative may tabulate crop yields by farm and season.
Example 1: A SACCO records loan disbursements (in Ksh thousands) by three groups over four months:
$$ L = \begin{bmatrix} 50 & 60 & 55 & 70 \\ 40 & 45 & 50 & 65 \\ 30 & 35 & 40 & 45 \end{bmatrix} $$
How many loan disbursement entries are recorded?
Answer:
Number of elements = \( 3 \times 4 = 12 \)
Example 2: A hotel tracks room bookings for 2 room types (Standard and Deluxe) over 3 months:
$$ B = \begin{bmatrix} 120 & 130 & 140 \\ 80 & 95 & 100 \end{bmatrix} $$
Identify the number of bookings for Deluxe rooms in the second month.
Answer: \( b_{22} = 95 \) bookings
Example 3: A university records number of students enrolled in 3 courses over 2 semesters:
$$ E = \begin{bmatrix} 200 & 220 \\ 180 & 190 \\ 150 & 160 \end{bmatrix} $$
Find the total enrollment for course 1.
Answer:
$$ 200 + 220 = 420 \text{ students} $$
$$ M = \begin{bmatrix} 12 & 15 & 20 \\ 10 & 18 & 25 \end{bmatrix} $$
find the element in the first row, third column. (2 marks)
$$ N = \begin{bmatrix} 5 & 8 \\ 7 & 3 \\ 6 & 4 \end{bmatrix} $$
(3 marks)
$$ P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
(2 marks)
$$ Y = \begin{bmatrix} 12 & 14 & 16 & 18 \\ 10 & 11 & 13 & 15 \\ 8 & 9 & 10 & 12 \end{bmatrix} $$
How many total data points are recorded? (2 marks)
$$ R = \begin{bmatrix} 25 & 30 & 35 \\ 20 & 22 & 28 \end{bmatrix} $$
What is the repayment amount by group 1 in month 2? (2 marks)
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Create a free accountThis chapter introduced the concept of matrices as a fundamental tool for organizing and manipulating data in rows and columns. It covered the basic operations of matrices including addition, subtraction, multiplication, and the impossibility of direct division, explaining how these operations are performed element-wise or via row-column multiplication. The methods for solving a 2 by 2 matrix system were explored, demonstrating how matrices can represent and solve simultaneous linear equations. The chapter then examined the determinant of a 2 by 2 matrix, showing its calculation and importance in matrix invertibility. Following this, the process of finding the inverse of a 2 by 2 matrix was detailed, highlighting its role in solving matrix equations. Finally, practical applications of matrices in business operations were discussed, illustrating how matrices simplify complex calculations in areas such as financial analysis and resource allocation. This comprehensive coverage equips students with essential matrix skills for various quantitative problems.
Given matrices \( A = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 4 \\ 6 & 2 \end{bmatrix} \), find \( A + B \). (2 marks)
A county referral hospital records patient admissions in two departments over two months as matrices \( X = \begin{bmatrix} 120 & 150 \\ 90 & 110 \end{bmatrix} \) and \( Y = \begin{bmatrix} 80 & 140 \\ 100 & 130 \end{bmatrix} \). Calculate the matrix representing the difference in admissions between the two months (\( X - Y \)). (2 marks)
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