Contextualization

Matrices are a fundamental concept in mathematics. They provide a neat and concise way to represent and manipulate a collection of numbers. While matrices may seem like an abstract mathematical concept, they actually have a wide range of real-world applications.

Matrices are used in many areas of science and engineering. For example, in physics, they are used to describe the motion of particles and the behavior of waves. In computer science, they are used in computer graphics, data analysis, and machine learning. In economics, they are used in input-output models and to solve systems of linear equations.

Introduction

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. For example, the matrix below has 2 rows and 3 columns, so it is a 2 by 3 matrix.

1  2  3
4  5  6

There are several different operations that can be performed on matrices, including addition, subtraction, and multiplication. These operations are defined in a way that makes matrices behave like numbers, which makes them very powerful tools in mathematics and its applications.

Matrix addition and subtraction are defined only for matrices of the same size. The sum (or difference) of two matrices is obtained by adding (or subtracting) the corresponding entries. For example, the sum of the two matrices below is:

1 + 2  3 + 4  5 + 6
7 + 8  9 + 10 11 + 12

= 3  7  11
  15 17 23

Matrix multiplication, on the other hand, is defined for any two matrices, as long as the number of columns in the first matrix is equal to the number of rows in the second matrix. The product of two matrices is obtained by multiplying the corresponding entries and then adding the results. For example, the product of the two matrices below is:

1 * 1 + 2 * 4 + 3 * 7  1 * 2 + 2 * 5 + 3 * 8  1 * 3 + 2 * 6 + 3 * 9
4 * 1 + 5 * 4 + 6 * 7  4 * 2 + 5 * 5 + 6 * 8  4 * 3 + 5 * 6 + 6 * 9

= 30  36  42
  66  81  96

Resources for Further Study

1. Khan Academy: Matrices
2. Math Is Fun: Matrices
3. Wolfram MathWorld: Matrix
4. Book: "Linear Algebra and Its Applications" by David C. Lay.
5. Book: "Matrix Analysis and Applied Linear Algebra" by Carl D. Meyer.

Practical Activity

Activity Title: Matrix Operations: Addition, Subtraction, and Multiplication

Objective of the Project:

The objective of this project is to strengthen your understanding of matrix operations (addition, subtraction, and multiplication) and how they can be applied to solve real-world problems.

Detailed Description of the Project:

In this group project, you will be creating a set of matrices and performing various operations on them. You will then use these operations to solve a real-world problem.

Necessary Materials:

• Pen and paper for brainstorming and sketching out ideas
• A calculator for performing the matrix operations
• Computer with internet access for research

Detailed Step-By-Step:

1. Formation of groups (3-5 students) and discussion: Form groups and start a discussion about matrices, their operations, and real-world applications.

2. Research: Each group should research and find at least two real-world scenarios where matrices are used.

3. Plan your matrices: Based on the scenarios found, plan and create two different matrices for each operation (addition, subtraction, and multiplication).

4. Perform the operations: Perform the matrix operations (addition, subtraction, and multiplication) for the created matrices.

5. Solve a Problem: Use these operations to solve a problem related to the real-world scenarios you found.

6. Document and Present: Document all the steps of your project and present your findings to the class.

Project Deliveries:

1. Written Document: A report of the project containing:

   • Introduction: Contextualize the theme, its relevance, and real-world application, as well as the objective of this project.
   • Development: Detail the theory behind matrix operations (addition, subtraction, and multiplication), explain the activity in detail, indicate the methodology used, and finally present and discuss the results.
   • Conclusion: Revisit the main points, state the learnings obtained, and the conclusions drawn about the project.
   • Bibliography: Indicate the sources you relied on to work on the project such as books, web pages, videos, etc.

2. Presentation: Each group will present their findings to the class. The presentation should include an explanation of the real-world scenarios, the matrices created, the operations performed, and the problem solved using the operations.

Note: The project is expected to take no more than two hours per participating student to complete and should be delivered within one week from the project assignment date.