Objectives (5 - 7 minutes)

1. Understanding the Inverse Matrix: The main objective is for students to understand the concept of an inverse matrix, its properties, and the importance of the determinant for its calculation. They should be able to identify an inverse matrix and how it is calculated.

2. Calculating the Inverse Matrix: Students should be able to calculate the inverse of a given square matrix. They will practice applying formulas and algorithms for calculating the inverse matrix.

3. Using Cofactors in Determinant Calculation: The third objective is for students to understand the use of cofactors in calculating the determinant of a matrix. They should be able to identify cofactors and apply them correctly in the determinant calculation.

Secondary Objectives:

• Practical Application: Students should be able to apply the concepts learned in solving practical problems.

• Development of Logical-Mathematical Thinking: Through the calculation of inverse matrices and determinants, students will develop logical-mathematical thinking skills.

Introduction (10 - 12 minutes)

1. Recalling Necessary Concepts: The teacher should start the lesson by recalling the concepts of matrix and determinants, as these are essential for understanding the lesson's topic. The difference between a square matrix and an inverse matrix should also be highlighted.

2. Problem Situations: The teacher can propose two problem situations to introduce the topic:

   • Situation 1: Imagine you have a matrix representing a system of linear equations. How can you use the inverse matrix to solve this system of equations?
   • Situation 2: Suppose you have a square matrix and want to know if it is invertible. What is the necessary condition for a matrix to be invertible?

3. Contextualization: The teacher should explain the importance of studying inverse matrices and determinants, highlighting their applications in various fields such as engineering, physics, economics, and computer science. For example, in engineering, inverse matrices are often used to solve systems of linear equations, while determinants are used to calculate areas and volumes.

4. Introduction to the Topic: To spark students' interest, the teacher can share some curiosities or interesting applications of the topic:

   • Curiosity 1: The idea of an inverse matrix dates back to Ancient Mathematics but was only formalized in the 19th century. It plays a fundamental role in many modern algorithms, including those used in computer graphics and cryptography.
   • Curiosity 2: The determinant of a matrix has an interesting geometric meaning. If the determinant is positive, the matrix represents a transformation that preserves orientation (i.e., does not invert the figure). If the determinant is negative, the matrix inverts the figure's orientation.

5. Gaining Attention: Finally, to gain students' attention, the teacher can present a challenge:

   • Challenge: "Did you know that matrices have an 'age'? The age of a matrix is the smallest number of times you need to multiply it by itself to obtain the identity matrix. For example, the age of the matrix [[1, 2], [3, 4]] is 2 because we need to multiply it by itself twice to obtain the identity matrix. Can you discover the age of other matrices?"

This is a topic that can spark students' curiosity and encourage them to explore more about the subject.

The teacher should conclude the Introduction by emphasizing the importance of the topic and encouraging students to actively participate in the lesson, asking questions and proposing solutions to the problems presented.

Development (20 - 25 minutes)

1. Developing the Concept of Inverse Matrix (7 - 10 minutes):

   • Definition of Inverse Matrix: The teacher should start by clearly explaining what an inverse matrix is, emphasizing that a matrix A has an inverse matrix if and only if the determinant of A is different from zero.
   • Calculating the Inverse Matrix: Next, the teacher should teach the formula for calculating the inverse matrix. They should demonstrate the process step by step, using simple examples. It is important to emphasize that calculating the inverse matrix involves using cofactors and the matrix's determinant.
   • Properties of the Inverse Matrix: The teacher should present the properties of the inverse matrix, such as the fact that the inverse of the inverse of a matrix is the matrix itself, and that the inverse matrix of a product of matrices is the product of the inverse matrices in reverse order.

2. Cofactors and Determinants (7 - 10 minutes):

   • Definition of Cofactor: The teacher should explain what cofactors are and how they are calculated. They should emphasize that the cofactor of an element aij of a matrix is the determinant of the submatrix resulting from excluding row i and column j.
   • Calculating the Determinant with Cofactors: The teacher should demonstrate how to calculate the determinant of a matrix using cofactors. They should show the process step by step, using simple examples. It is important to highlight that calculating the determinant using cofactors is a recursive process that involves calculating determinants of smaller submatrices.
   • Applications of Determinants: The teacher should present some applications of determinants, such as determining the existence of an inverse matrix, calculating the area of a parallelogram or the volume of a parallelepiped, and determining the orientation of a figure after a linear transformation.

3. Practical Activities (6 - 8 minutes):

   • Solving Exercises: The teacher should propose some exercises for students to solve, both for calculating the inverse matrix and the determinant with cofactors. They should circulate around the room, assisting students who are having difficulty and clarifying doubts.
   • Group Discussion: After solving the exercises, the teacher should promote a group discussion so that students can share their solving strategies and clarify doubts. This is a great opportunity for students to learn from each other and for the teacher to assess the class's level of understanding.
   • Feedback and Error Correction: The teacher should correct the exercises together with the class, providing feedback on the answers and clarifying any errors or misunderstandings. It is important for students to understand not only the correct result but also the solving process.

The teacher should ensure that all students are following the lesson and understanding the concepts, adjusting the lesson's pace as necessary. They should encourage students to ask questions and actively participate in discussions and activities.

Return (8 - 10 minutes)

1. Reviewing Concepts (3 - 4 minutes):

   • The teacher should start the Return by reviewing the fundamental concepts covered in the lesson. They can ask direct questions to students or ask them to describe in their own words what they understood about the inverse matrix, determinants, and cofactors.
   • The teacher should reinforce the importance of each concept and how they relate. For example, the inverse matrix can only be calculated if the matrix's determinant is different from zero. Cofactors, in turn, are used in the determinant calculation.
   • It is important for the teacher to check if all students have understood these concepts, clarifying any doubts that may arise.

2. Connection with Practice (2 - 3 minutes):

   • The teacher should then connect the presented theory with the practical applications discussed in the Introduction. They can recall examples of how inverse matrices and determinants are used in solving systems of linear equations, calculating areas and volumes, and determining the orientation of a figure after a linear transformation.
   • The teacher can propose a reflection exercise, asking students to think of other practical situations where these concepts can be applied. For example, how inverse matrices can be useful in computer programming, or how determinants can be used in physics to determine if a force applied to an object will alter its shape.

3. Reflection on Learning (2 - 3 minutes):

   • The teacher should encourage students to reflect on what they learned in the lesson. They can ask questions like: "What was the most important concept you learned today?" or "What questions remain unanswered?"
   • The teacher should give a minute for students to think and then ask some of them to share their answers with the class. It is important for the teacher to value all contributions and for students to feel comfortable expressing their doubts and difficulties.
   • The teacher can make a brief assessment of learning, asking students to raise their hands if they feel they understood most of the concepts presented. This can help the teacher identify any areas that need to be reviewed or reinforced in future lessons.

4. Closure (1 minute):

   • To close the lesson, the teacher should summarize the main points discussed and reinforce the importance of continuous study and practice for understanding these concepts. They should thank the students for their participation and effort and encourage them to keep striving in their studies.

This Return is essential for consolidating students' learning and for the teacher to assess the lesson's effectiveness. The teacher should be attentive to students' doubts and difficulties and adjust the planning of future lessons as necessary.

Conclusion (5 - 7 minutes)

1. Recapitulation of Main Contents (2 - 3 minutes):

   • The teacher should start the Conclusion by recapitulating the main points covered in the lesson, such as the definition and calculation of the inverse matrix, the importance of cofactors and the determinant, and their practical applications.
   • They should reinforce the most important concepts and ensure that students have understood the theory behind the calculation of the inverse matrix and determinants.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes):

   • Next, the teacher should highlight how the lesson connected theory, practice, and applications. They can, for example, mention how practical exercises helped students better understand theoretical concepts and how practical applications demonstrated the relevance of these concepts.
   • The teacher should reinforce that Mathematics is not just a set of rules and formulas but a powerful tool for understanding and solving real-world problems.

3. Extra Materials (1 minute):

   • The teacher can suggest some additional study materials for students who want to deepen their knowledge on the topic. This can include books, websites, videos, and online exercises.
   • They can, for example, recommend a Khan Academy video on calculating the inverse matrix and determinants or a math practice website with interactive exercises.

4. Relevance of the Topic (1 - 2 minutes):

   • Finally, the teacher should emphasize the importance of the topic for everyday life and other areas of knowledge. They can, for example, mention how calculating the inverse matrix is used in solving systems of linear equations, a common problem in many areas of science and engineering.
   • The teacher can also reinforce how understanding determinants can help grasp important concepts in Geometry, Physics, and other disciplines.
   • They should conclude the lesson by reinforcing that, despite being challenging, the study of matrices and determinants is fundamental for developing logical-mathematical thinking skills and understanding many other mathematical concepts.

This Conclusion is essential for consolidating students' learning, reinforcing the topic's relevance, and motivating students to continue studying and exploring the subject.