Contextualization

Mathematics is a fundamental language for expressing and solving complex real-world problems. Among the numerous concepts in this vast discipline, determinants emerge as a crucial tool for solving systems of linear equations, among other applications. This project will focus specifically on determinants of a matrix with one row and one column, also known as '1x1 Determinant'.

Introduction

The concept of determinants dates back to the 3rd century BC in China and gained remarkable development from the 18th century onwards, with the contribution of mathematicians such as Leibniz and Laplace. In its simplest form, the 1x1 determinant is essentially the element of the matrix itself. Let A be a 1x1 matrix, containing only one element 'a', the determinant of this matrix is the value 'a' itself. This concept, although simple, is the basis for understanding determinants of larger order matrices. It is from the idea of this basic determinant that we can abstract to determinants of 2x2, 3x3, and so on.

The importance of the 1x1 matrix and its determinant, however simple they may seem, is immeasurable. It is the basis for understanding the entire universe of matrices and determinants. It is from it that we build and understand the properties of matrices of higher orders. Furthermore, the direct application of the 1x1 determinant is in simple linear equations, which can be found in various contexts, from resource allocation in a company to solving physical problems.

Relevance

In this project, you will discover a variety of surprising examples and applications of 1x1 matrices and their determinants, including how they are used in scientific research, economic decision-making, and engineering. For example, researchers in quantum physics use determinants to describe properties of subatomic particles, while economists use determinants to model consumer behavior and make economic predictions.

Practical Activity

Activity Title: 'Determinant Detective'

Project Objective:

This project aims to unveil the world of determinants applied to solving real-world problems. To do so, the group must create a 'case' that will be solved through the use of 1x1 matrices and their determinants.

Project Description:

Each group must create a narrative in which 1x1 determinants are the key to the solution. This narrative can be a business challenge, a scientific puzzle, a logistical problem, among others. The narrative should involve at least four different concepts or applications of determinants, as well as explore the intersection between Mathematics and another discipline of interest to the group.

Required Materials:

• Whiteboard or flipchart for project presentation.
• Colored markers.
• Computer with internet access for research and narrative development.
• Text editing software for creating the project report.

Project Step-by-Step:

1. Form groups of 3 to 5 students.
2. Choose the second discipline that will be addressed together with mathematics.
3. Brainstorm and decide on the narrative: Who are the characters? What is the problem that needs to be solved? How will the 1x1 determinant be used in the solution?
4. Research the applications of the 1x1 determinant in the chosen narrative.
5. Create a script, highlighting four different concepts or applications of determinants in your story.
6. Develop a 15-20 minute presentation to tell the narrative and explain the concepts of determinants involved.
7. In parallel with the presentation, develop a detailed project report.

Project Delivery:

In addition to the presentation, each group must deliver a written report describing the narrative, the mathematical concepts involved, the interdisciplinary applications, and the conclusions. The report should be structured as follows:

1. Introduction: In this section, the group must contextualize the created narrative, mentioning the relevance of mathematics and specifically of 1x1 determinants for the proposed solution. The group must also explain the reason for choosing the second discipline and how it integrates into the narrative.

2. Development: In this section, the group must detail the narrative, explaining step by step how the 1x1 determinants were used to solve the problem. The group must also detail the theory behind 1x1 determinants and explain how they were applied at each stage of the solution.

3. Conclusions: In this section, the group must discuss the main learnings obtained during the project, emphasize the importance of teamwork, and comment on the interdisciplinarity of the project.

4. Bibliography: The group must cite all resources used during the project: books, websites, videos, among others.

This project has an estimated duration of 12 hours per student and should be developed over a few weeks.