Compiled by: Samuel Maina
Cooperative management professionals often deal with data organized in tabular form, such as member contributions, loan amounts, and repayment schedules. Matrices provide a structured way to represent, analyze, and manipulate such data efficiently. Mastery of matrix concepts and operations enhances decision-making, financial analysis, and record-keeping in cooperative societies.
Matrices are rectangular arrays of numbers arranged in rows and columns that represent data or coefficients in linear algebraic equations. In cooperative management, matrices can model financial transactions, member data, or resource allocation. A matrix is denoted by a capital letter, such as \( A \), and its elements by \( a_{ij} \), where \( i \) is the row number and \( j \) is the column number. The size of a matrix is given by the number of rows \( m \) and columns \( n \), written as an \( m \times n \) matrix.
A matrix \( A \) with 2 rows and 3 columns looks like:
$$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} $$
Matrices can be classified by size (row and column count) and type (square, zero, diagonal, identity, etc.), each useful for different applications in cooperative data analysis.
Matrix dimensions are critical to understanding how matrices can be combined or manipulated. The dimension of a matrix is expressed as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Each element in the matrix is identified by its position \( (i, j) \), where \( i \) is the row index and \( j \) is the column index.
$$ A = [a_{ij}], \quad i = 1, 2, ..., m; \quad j = 1, 2, ..., n $$
Example 1: A cooperative has a member loan matrix \( L \) with 3 rows (members) and 2 columns (loan types). Write the dimension of matrix \( L \).
Given: Rows \( m = 3 \), Columns \( n = 2 \)
Answer: Matrix \( L \) is a \( 3 \times 2 \) matrix.
Example 2: Identify the element in the 2nd row and 1st column of matrix
$$ M = \begin{bmatrix} 5 & 8 \\ 12 & 7 \\ 9 & 4 \end{bmatrix} $$
Given: \( i=2 \), \( j=1 \)
Element \( a_{21} = 12 \)
Answer: The element is 12.
Example 3: A matrix \( N \) has 4 rows and 4 columns. What type of matrix is it if the diagonal elements are all 1 and the rest are 0?
Given: \( 4 \times 4 \) matrix with diagonal 1s, others 0.
Answer: \( N \) is an identity matrix.
Cooperative data often involves specific matrix types such as square, diagonal, zero, and identity matrices. Understanding these types aids in selecting appropriate matrix operations.
Example 1: Determine if matrix
$$ Q = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} $$
is diagonal.
Given: Elements off the diagonal are zero.
Answer: \( Q \) is a diagonal matrix.
Example 2: Identify if matrix
$$ Z = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$
is zero matrix.
Answer: \( Z \) is a zero matrix.
Example 3: Verify if matrix
$$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
is identity matrix.
Answer: \( I \) is an identity matrix.
Matrices can compactly represent cooperative data such as member contributions, loan balances, and repayment schedules. Each row can represent a member, and each column a data variable.
Example: Member contributions over three months:
$$ C = \begin{bmatrix} 5000 & 4000 & 5500 \\ 3000 & 3500 & 4000 \\ 4500 & 4700 & 4800 \end{bmatrix} $$
Rows: Members 1, 2, 3; Columns: Month 1, 2, 3
Example 1: Calculate total contribution for member 2.
Given matrix \( C \) above.
Formula:
$$ \text{Total} = \sum_{j=1}^3 c_{2j} $$
$$ = 3000 + 3500 + 4000 $$
$$ = 10500 \text{ Ksh} $$
Answer: Member 2 contributed 10,500 Ksh.
Example 2: Find the contribution in month 3 by all members.
$$ \sum_{i=1}^3 c_{i3} = 5500 + 4000 + 4800 = 14300 \text{ Ksh} $$
Answer: Total contribution in month 3 is 14,300 Ksh.
Example 3: Find the average monthly contribution for member 1.
$$ \frac{\sum_{j=1}^3 c_{1j}}{3} = \frac{5000 + 4000 + 5500}{3} = \frac{14500}{3} = 4833.33 \text{ Ksh} $$
Answer: Average contribution is 4,833.33 Ksh.
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Create a free accountThis chapter introduced the fundamental concept of matrices, explaining their structure and significance in organizing data. It then detailed the basic operations of matrices, including addition, subtraction, division, and multiplication, highlighting the rules that govern each operation. The process of solving a 2 by 2 matrix was explored, providing a foundation for handling simple matrix equations. The chapter further examined the calculation of determinants for a 2 by 2 matrix, an essential step in understanding matrix properties. Building on this, the method for finding the inverse of a 2 by 2 matrix was presented, showing how it can be used to solve matrix equations. Finally, the chapter demonstrated practical applications of matrices in solving business operations, illustrating their value in managing and analyzing business data effectively.
Given matrices \( A = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \), find \( A + B \). (2 marks)
A cooperative has two branches with monthly sales (in Ksh thousands) represented by matrices: \( C = \begin{bmatrix} 20 & 30 \\ 15 & 25 \end{bmatrix} \) for January and \( D = \begin{bmatrix} 10 & 20 \\ 5 & 15 \end{bmatrix} \) for February. Find the decrease in sales by calculating \( C - D \). (3 marks)