Introduction

Relevance of the Topic

Determinants are a fundamental tool in the study of linear algebra, being one of the first 'cornerstones' in this discipline. They have applications in various areas, from solving systems of linear equations to geometry, physics, and economics. Understanding their properties is vital for the efficient and correct resolution of many mathematical and real-world problems.

Contextualization

The study of determinants is introduced after becoming familiar with basic matrix operations. With a deepening in this subject, the goal is for students to not only understand the definition of a determinant but also master its properties. The properties of determinants have numerous applications, especially when it comes to operations with matrices, such as matrix inversion and solving linear systems. Familiarity with these properties will provide a solid foundation for the study of more advanced topics in mathematics, such as linear transformations and vector spaces.

Theoretical Development

Components

• Rule of Null Product: Every determinant of a square matrix is null if the matrix has a null row (or column).

• Row (or Column) Interchange: If the rows (or columns) of a matrix are interchanged, the absolute value of its determinant remains the same, but the determinant's sign may change, meaning the determinant can become the opposite of the original value.

• Multiplication of a Single Row (or Column) by a Scalar: If a row (or column) of a matrix is multiplied by a scalar k, the absolute value of the determinant is multiplied by k, meaning the determinant becomes the original determinant multiplied by k.

• Addition of a Multiple of One Row (or Column) to Another Row (or Column): If a row (or column) of a matrix is replaced by the sum of that row (or column) and a multiple of another row (or column), the absolute value of the determinant remains unchanged.

Key Terms

• Determinant: It is a number that can be calculated for a square matrix. In a 2x2 matrix, the determinant is calculated by subtracting the product of the elements of the secondary diagonal from the product of the elements of the main diagonal. In a 3x3 matrix, the determinant is calculated by 'Sarrus' rule.

• Square Matrix: A matrix is said to be square when the number of rows is equal to the number of columns.

Examples and Cases

• Rule of Null Product: Given the matrix A = [1 2 3; 0 0 0; 4 5 6], note that the second row is null. Therefore, the determinant of A is zero.

• Row (or Column) Interchange: Let matrix B = [1 2; 3 4]. The determinant of B is -2. However, if we swap the rows, the matrix becomes C = [3 4; 1 2]. The determinant of C is now 2, which is the opposite of -2.

• Multiplication of a Single Row (or Column) by a Scalar: Consider matrix D = [1 2; 3 4]. The determinant of D is -2. Now, if we multiply the second row by -2, we get matrix E = [1 2; -6 -8]. The determinant of E is -16, which is -2 multiplied by -8.

• Addition of a Multiple of One Row (or Column) to Another Row (or Column): Let's take matrix F = [1 2; 3 4]. The determinant of F is -2. If we add the second row to the first, the matrix becomes G = [4 6; 3 4]. The determinant of G is still -2.

Detailed Summary

Key Points

• Importance of Determinants: Determinants are a central and versatile tool in mathematics. They assist in many topics, from solving linear systems to geometry, physics, and economics.

• Rule of Null Product: This property of determinants establishes that if a square matrix has a null row (or column), its determinant will be zero. This rule is fundamental for identifying singular matrices, a term used to describe matrices that cannot be inverted.

• Row (or Column) Interchange: Interchanging rows or columns in a matrix does not change the absolute value of the determinant, but it can change its sign, resulting in a negative value if the swap is made.

• Multiplication of a Single Row (or Column) by a Scalar: This property states that if we multiply a row (or column) of a matrix by the scalar k, the resulting determinant will be the original determinant multiplied by k. This rule is often used to simplify calculations or to identify common matrices in various situations.

• Addition of a Multiple of One Row (or Column) to Another Row (or Column): This property establishes that if we add a multiple of one row (or column) to another row (or column) of the matrix, the absolute value of the determinant does not change. This property is useful in matrix manipulations to obtain a simpler determinant value.

Conclusions

• The study of determinant properties is vital to deepen the knowledge about matrices, as they provide information about the matrix that cannot be obtained just by observing its elements.

• The properties of determinants can be used to simplify calculations or solve problems where matrix manipulation is required.

Exercises

1. Calculate the determinant of the following matrix and verify the Rule of Null Product:

   • A = [1 2 3; 0 0 0; 4 5 6]

2. Find the value of the determinant of matrix B = [1 -2 -3; 4 5 -6; 7 8 9] by making changes and apply the studied rules to determine which property was used at each step.

3. Given matrix C = [1 2; 3 4], perform the following operations and calculate the determinant at each step to verify the studied properties:

   • Multiply the first row by 2 and subtract from the second row
   • Multiply the first column by -3 and add to the second column.