Contextualization

Determinants are one of the most fundamental concepts in linear algebra. They are a special function that assigns to each square matrix a scalar. In this project, we will be focusing on the simplest form of determinant, the 1x1 determinant. This term might sound daunting, but it simply means the determinant of a single number, or matrix with only one row and one column.

Determinants have a wide range of applications in mathematics, physics, computer science, and many other fields. They are used to solve systems of linear equations, find the area of a parallelogram, and even to determine the invertibility of a matrix. They are an essential component of linear algebra, a branch of mathematics that studies linear equations and their representations in vector spaces and through matrices.

Let's consider the following example to understand determinants better:

|7|  => The determinant of this 1x1 matrix is 7.

While this might seem like a trivial example, understanding the concept of determinants in this simple form will set the foundation for tackling more complex determinants and their applications.

Importance

The study of determinants is essential in understanding the properties and behavior of matrices, which are widely used in various fields of science, engineering, and computing. They are particularly important in linear algebra, a key subject in mathematics that underpins many other areas of study.

Moreover, determinants have practical applications in areas such as physics, where they are used to calculate the moment of inertia of an object, and even in economics, where they are used to model and solve problems related to supply and demand. By understanding determinants, you are equipping yourself with a powerful mathematical tool that has real-world applications.

Resources

To delve deeper into the concept of determinants, you can refer to the following resources:

1. Khan Academy: Determinants: This video tutorial provides a great introduction to determinants, including 1x1 determinants.

2. Math is Fun: Determinants: This website offers a clear and concise explanation of determinants, with helpful visuals.

3. Brilliant: Matrices and Determinants: This resource not only covers the basics of determinants but also provides more advanced topics for those who want to go deeper.

4. YouTube: The Determinant: A video lecture series from MIT OpenCourseWare, this is a more in-depth look at determinants, but it can provide a solid understanding for those who are ready for a challenge.

Remember, understanding the concept is just the first step. To truly master determinants, it's important to practice solving problems and applying the theory to real-world examples. Good luck!

Practical Activity

Activity Title: "The Power of One"

Objective of the Project:

The aim of this project is to understand the concept of determinants, with a specific focus on 1x1 determinants. Students will learn how to calculate the determinant of a 1x1 matrix, understand its significance, and apply this knowledge to solve problems.

Detailed Description of the Project:

In this project, each group of students will create a presentation that explains and demonstrates the concept of 1x1 determinants. The presentation should include a theoretical explanation of 1x1 determinants, real-world examples, and a step-by-step guide on how to calculate the determinant of a 1x1 matrix.

Each group will also need to conduct a practical experiment where they use 1x1 determinants to solve a problem. This problem should be a real-world application of 1x1 determinants, and the solution should be presented using the knowledge gained from the theoretical part of the project.

Necessary Materials:

• Pen and paper for calculations and brainstorming
• Computers with internet access for research and presentation creation

Detailed Step-by-Step for Carrying Out the Activity:

1. Formation of Groups and Allocation of Tasks (1 hour): Divide the students into groups of 3-5. Each group should have a leader, a researcher, a writer, and a presenter. The leader will be responsible for coordinating the tasks, the researcher for finding relevant information, the writer for drafting the report, and the presenter for creating the presentation and delivering it.

2. Research (1-2 hours): The researcher should start by understanding the concept of 1x1 determinants using the provided resources. They should then find real-world examples of 1x1 determinants and how they are used. The leader and writer should assist the researcher in this task.

3. Presentation Planning (1 hour): Once the research is complete, the group should plan their presentation. This should include deciding on the structure, assigning tasks for creating slides and content, and setting a timeline for completion.

4. Presentation Creation (2-3 hours): The presenter and writer should work together to create the presentation based on the research and planning. This should include a theoretical explanation, real-world examples, and a step-by-step guide on how to calculate the determinant of a 1x1 matrix. The presentation should be clear, engaging, and easy to understand.

5. Practical Experiment (1-2 hours): The group should come up with a real-world problem that can be solved using 1x1 determinants. They should then use the knowledge from their research and presentation to solve the problem. The solution to the problem should be presented as part of the final presentation.

6. Review and Finalization (1 hour): The entire group should review the presentation, make any necessary changes, and finalize it. The report should also be reviewed and edited if necessary.

7. Presentation and Report Submission (30 minutes): The group should present their work to the class and submit their report.

Deliverables:

1. Presentation: Each group should create a PowerPoint presentation, Prezi, or any other suitable format that explains the concept of 1x1 determinants, provides real-world examples, and demonstrates a solution to a real-world problem using 1x1 determinants.

2. Report: The report should be divided into four main sections: Introduction, Development, Conclusions, and Used Resources. The Introduction should contextualize the theme, its relevance, real-world application, and the objective of the project. The Development section should detail the theory of 1x1 determinants, explain the activity in detail, indicate the methodology used, and finally present and discuss the obtained results. The Conclusion should revisit the main points of the report, explicitly stating the learnings obtained and the conclusions drawn about the project. Finally, the Used Resources section should list all the sources that were used to work on the project such as books, web pages, videos, etc.

The project's duration is estimated to be about 5-8 hours per student. It is expected that each group will complete the project within one week.