Contextualization

The study of Determinants is a fundamental topic when it comes to mathematics. They play a unique role in calculations of areas, volumes, in the solution of systems of linear equations, and are very present in the universe of pure and applied Mathematics. Our focus here will be on the determinant of a 2x2 matrix, an essential topic that will allow the understanding of these more complex applications later on.

The theory of determinants dates back centuries, but its use and formal definition as we know it today were developed during the 18th century. A determinant is a special function that involves a simple calculation from the elements of the matrix as a whole, and not just the product of its individual elements.

Introduction

The most well-known application of the 2x2 determinant is in the solution of systems of linear equations. This is an area of study that has practical applications in almost every field - from engineering to economics, through physics, biology, and computer science. The determinant helps us verify whether a system of equations has a solution or not, which can be crucial for solving a real problem.

Furthermore, the determinant is important for calculating the area of a triangle in the Cartesian plane, thus being fundamental for applications in fields such as architecture, design, and physical geography.

The resources we suggest for a more in-depth study of the topic are the book "Linear Algebra with Applications" by Anton Howar, which provides a complete and detailed approach to the subject, and the website Só Matemática (https://www.somatematica.com.br/emedio/sistemas/sistemas5.php), which contains step-by-step examples of how to deal with determinants of 2x2 matrices and their applications in systems of equations.

Practical Activity

Activity Title: Unraveling the Determinant: Systems Solution and Triangle Construction

Project Objective

This project aims to explore the concept of determinants of a 2x2 matrix and its applications in solving systems of linear equations and in the construction of triangles in a fictitious urban planning.

Detailed Project Description

Students, divided into groups of 3 to 5, must first study the theory of determinants of a 2x2 matrix, using the suggested resources mentioned earlier and any other materials they find relevant. They must then apply this knowledge in solving some systems of equations previously prepared by the teacher.

After solving these systems, the groups must now use the determinants of 2x2 matrices in the design of a fictitious urbanization project. They must draw a map of a city, including squares and parks in the shape of triangles. The area of these triangles must be calculated using determinants.

Next, they must produce a report explaining the entire process, from the theory of determinants to the application of the same in practical problems.

Required Materials

• Book "Linear Algebra with Applications" by Anton Howar
• Access to the website Só Matemática
• Paper and pencil for drawing the map
• Calculator

Detailed Step-by-Step for Carrying Out the Activity

1. Study the theory of 2x2 determinants using the indicated materials.
2. Solve the systems of equations proposed by the teacher using determinants and verify if the solution was correct.
3. Draw a map of a fictitious city including urban areas (represented by triangles).
4. Calculate the area of the triangles using determinants.
5. Write a detailed report on the process, including the theory of determinants, the resolution of the systems of equations, and the application of determinants in the urban project.

Project Delivery

Project deliveries include the drawn map of the city, with the areas of the triangles labeled, and the written report. The report should be structured in the following topics:

• Introduction: Contextualization of the theme, its relevance and application in the real world, and the objective of this project.
• Development: Explanation of the theory of determinants, detailing how the systems of equations were solved, description of the urbanization project, with special emphasis on how the areas of the triangles were calculated.
• Conclusions: Recap of the main points, explanation of the learnings obtained, and conclusions drawn about the project.
• Bibliography: Indication of the sources they relied on to work on the project such as books, web pages, videos, etc.

These activities offer students the opportunity to deepen their knowledge in determinants while developing important skills such as time management, problem-solving, creative thinking, and proactivity.