INTRODUCTION

Relevance of the Topic

The calculation of determinants is a fundamental part of Linear Algebra. It provides crucial information about the matrix, especially regarding its invertibility. Specifically, the 3x3 determinant is particularly important as it acts as a building block for understanding larger determinants and systems of linear equations. Without a solid understanding of 3x3 determinant calculation, some of the more advanced concepts in Mathematics, such as vector spaces and linear transformations, may become inaccessible.

Contextualization

Following the mathematics curriculum, understanding and calculating determinants of a 3x3 matrix usually comes after studying systems of linear equations and matrix multiplication. Therefore, this section is a natural transition, as it combines with these previous topics to deepen the understanding of the field of study. Moreover, understanding the 3x3 determinant is essential for solving systems of linear equations with three variables, a crucial aspect of Linear Algebra used in many fields, including natural sciences, engineering, statistics, and economics.

THEORETICAL DEVELOPMENT

Components

• 3x3 Matrix: A regular two-dimensional matrix consisting of three rows and three columns. Each element (a, b) represents the value in row a, column b. Its structure is of fundamental importance for the calculation of the 3x3 determinant.

• Co-factor: It is a real number associated with each element of a matrix. The co-factor of an element is the determinant of the submatrix obtained by removing its row and column. In the calculation of 3x3 determinants, the co-factor plays a vital role in Leibniz's formula.

• Leibniz's Formula: A standard formula for calculating the determinant of a 3x3 matrix. It is based on the summation of products of matrix elements with their respective co-factor. Leibniz's formula is an essential component of the study of 3x3 determinants.

• Determinant: The determinant is a real number associated with each square matrix. In the case of 3x3 matrices, the determinant provides information about the linear transformation represented by the matrix. For example, it is zero if and only if the matrix is not invertible, and its value changes with changes in scale and rotation of the coordinate system.

Key Terms

• 3x3 Determinant: The determinant of a 3x3 matrix is the result obtained by using Leibniz's formula to calculate the summation of products of matrix elements with their respective co-factors. This is a numerical value that provides a lot of information about the matrix and is fundamental to many aspects of Mathematics.

• Invertible: A matrix is called "invertible" or "non-singular" if its determinant is different from zero. This is a crucial property that has significant implications in Linear Algebra.

• Systems of Linear Equations: A direct application of determinant calculation is the resolution of systems of linear equations. The solutions to such systems are closely linked to the value of the determinant of the matrix of coefficients.

Examples and Cases

• Change of Basis Matrix: In Geometry, change of basis matrices are used to convert the coordinates of a vector from one reference system (or base) to another. The determinant of the change of basis matrix is particularly important, as it provides information on how the area or volume of an object (determined by that vector) changes when viewed in different coordinate systems.

• Resolution of a System of Linear Equations: When solving a system of linear equations, the matrix of coefficients is usually converted into its augmented matrix and the value of the determinant is calculated. If the determinant is zero, the system of equations does not have a single solution (i.e., it is indeterminate or impossible). If the determinant is different from zero, the system has exactly one solution.

These examples, along with the components and key terms discussed, establish the foundation for a solid understanding of the topic of 3x3 determinants.

DETAILED SUMMARY

Relevant Points

• 3x3 Matrix: The 3x3 matrix is a fundamental structure in the calculation of determinants. It has three rows and three columns, an arrangement that allows the application of Leibniz's formula for its calculation.

• Co-factor and Leibniz's Formula: The calculation of 3x3 determinants involves Leibniz's formula, which uses co-factors to calculate the result. The co-factor of an element is the determinant of the submatrix obtained by removing its row and column. The use of co-factors in Leibniz's formula allows for the efficient computation of the 3x3 determinant.

• Determinant: The determinant of a 3x3 matrix is a real number that provides crucial information about the matrix. It is zero if and only if the matrix is not invertible.

• Invertibility: The invertibility of a matrix, a fundamental concept in Linear Algebra, is directly related to the value of the determinant. A matrix is invertible if and only if its determinant is different from zero.

• Practical Applications: The calculation of 3x3 determinants has applications in various areas, including Geometry and solving systems of linear equations.

Conclusions

• The ability to calculate determinants of 3x3 matrices is essential for understanding and applying fundamental and advanced concepts of linear algebra.
• The value of the determinant provides essential information about the matrix, including its invertibility, changes in scale and volume, and more.
• The calculation of the 3x3 determinant depends on the understanding and application of Leibniz's formula and co-factors, demonstrating the interconnection of matrix and determinant concepts.

Suggested Exercises

1. Calculate the determinant of A: A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Analyze the result obtained and check if there is any geometric interpretation for it.

2. Check invertibility: For matrices B = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] and C = [[1, 2, 3], [0, 1, 4], [5, 6, 7]], calculate the determinant and check if the matrices are invertible.

3. System of Linear Equations: Given the system of linear equations below, calculate the determinant of the matrix of coefficients and check if the system has a single solution.

   • x + 2y + 3z = 10
   • 4x + 5y + 6z = 20
   • 7x + 8y + 9z = 30