Contextualization

Determinants are a fundamental tool in Mathematics, especially when it comes to linear systems and linear transformations. But what is a determinant? In simple terms, it is a special number that can be calculated from a square matrix. The importance of the determinant lies in its multiple applications, such as deciding whether a system of linear equations has a unique solution, in matrix inversion processes, or even in areas such as analytical geometry and linear optimization.

The concept of a determinant, although it may seem abstract at first, actually permeates several practical issues. It is present when we calculate areas and volumes, in the analysis of system stability, and even in applied sciences such as economics, physics, and engineering. For example, in economics, determinants help in the study of demand and supply systems and in the analysis of equilibrium points.

To take a step forward and bring the determinant to practical life, let's explore the properties of determinants. It is these properties that make calculating determinants faster and more understandable, and they are fundamental to avoiding common errors in their application. Properties such as swapping rows changing the sign of the determinant, or the fact that the determinant of a matrix with a row of zeros is zero, are just the tip of the iceberg of this fascinating topic.

Reliable Resources to Delve into the Topic

To delve deeper into the topic of determinants and their properties, I suggest the following resources, all in Portuguese:

• Book "Fundamentals of Elementary Mathematics: Matrices, Determinants, and Systems" - Gelson Iezzi: a classic reference that offers an in-depth approach to the subject with practical exercises.
• "Matemática Rio" Channel on YouTube: has video lessons that approach the subject of determinants in a didactic and accessible way.
• "Matrices and Determinants" on the Só Matemática website (https://www.somatematica.com.br): offers a theoretical explanation along with examples that facilitate the understanding of the topic.
• Khan Academy (https://pt.khanacademy.org): offers interactive exercises and video lessons that cover a wide range of topics in mathematics, including matrices and determinants.

These resources should be used as a basis for the theoretical and practical understanding of the topic, and will serve to prepare students for the activity that will be proposed later. In addition, they stimulate debate and research, essential for the development of critical thinking and for the construction of knowledge.

Practical Activity

Activity Title

"Determinant Operation: Discovering and Applying Properties"

Project Objective

The project aims to engage students in understanding and applying the properties of determinants, stimulating teamwork, logical reasoning, and the ability to solve complex problems.

Detailed Project Description

Students will form groups of 3 to 5 members and conduct a practical investigation on the properties of determinants, as well as apply them in specific scenarios and create a didactic presentation on the subject.

Materials Required

• Graph paper or cardboard
• Colored pens and markers
• Internet access for research
• Graphing calculators or mathematical software (such as GeoGebra or MATLAB)
• Study guides or math textbooks

Detailed Step by Step

Week 1: Research and Theoretical Deep Dive

Each group should initially study and delve into the properties of determinants. The group should divide the topics among the members and each one will be responsible for becoming a "specialist" in one of the properties. The research should be documented to contribute to the 'Development' section of the report.

Weeks 2 and 3: Planning and Execution of the Practical Activity

The groups should plan and execute a series of practical activities that demonstrate the properties of determinants. Some suggestions include:

• Create a giant square matrix on graph paper and calculate its determinant using properties such as a row or column of zeros, scalar multiplication property, among others.
• Use software to simulate and visualize the effect of changes in determinants when applying elementary operations on rows or columns.
• Develop a small applied project that uses the concept of a determinant to solve a practical problem, such as the analysis of a system of linear equations in a real-world context (e.g., production optimization).

Week 4: Compilation of Results and Preparation of the Presentation

The groups should compile the results of their practical activities and prepare a presentation that demonstrates what they have learned. The presentation should include practical demonstrations, results obtained, and a clear explanation of the properties of the determinants used.

Project Deliverables

At the end of the month, each group must deliver:

1. Written Document: This report should contain the four main sections:

   • Introduction: contextualization and objective of the project.
   • Development: theoretical explanation of the properties of determinants and details of the methodology of the practical activities carried out.
   • Conclusions: synthesis of the learning and results, highlighting the relevance of the properties of the determinants.
   • Bibliography: list of all sources used.

2. Presentation: A visual presentation of the results and properties worked on, in a didactic and creative way.

Writing the Written Document

The group should write the written document as follows:

• Introduction: Contextualize the theme and explain the relevance of understanding the properties of determinants. Also present the objective of the didactic project carried out by the group.
• Development: Detail the properties of determinants that were studied, explain how the practical activity was planned and executed, and what the results observed were. Describe the participation of each group member and the difficulties encountered.
• Conclusions: Review the main points of the work, reflect on the learning and discoveries that emerged during the project, and how the determinant property was applied.
• Bibliography: List all sources that were consulted to support the theory and practice, including books, articles, videos, and websites.

The written document is an opportunity to show not only what was learned, but how it was learned, and should reflect the collaboration and interdisciplinarity of the practical activity.