Introduction

Relevance of the Theme

The calculation of determinants is one of the fundamental tools in the study of linear algebra. It has a series of practical and theoretical applications, being widely used in various areas of mathematics, engineering, and sciences. The 1x1 determinant is a special case that helps us understand in a broader and more intuitive way the calculation of determinants. By mastering the concept of 1x1 determinant and its properties, you will be strengthening your foundation for the study of determinants of higher order.

Contextualization

In the broader context of the high school mathematics curriculum, the study of determinants is often carried out as an extension of the study of matrices. After understanding how a matrix is created, the next step is to learn how to calculate its determinant. Furthermore, understanding determinants is crucial for the study of other topics, such as linear systems and linear transformations. This topic is therefore at the intersection point of several fundamental concepts in mathematics.

Theoretical Development

Components

• 1x1 Determinants: We define the determinant of a 1x1 matrix (i.e., a matrix with only one element) as the element itself. That is, if the matrix A = [a], the determinant of A is given by |A| = a.

• Matrices: Matrices are rectangular arrangements of numbers, symbols, or expressions, organized in rows and columns. The element a_ij is located in the i-th row and j-th column. For example, in the matrix A = [ [1, 2], [3, 4] ], we have a_12 = 2.

• nxn Determinants: For the calculation of determinants of order greater than 1, the process is more complex. It is necessary to perform a cofactor expansion, which is the sum of the products of the matrix elements by their respective cofactors.

Key Terms

• Determinant: It is a scalar value that can be calculated from the elements of a square matrix. It is important to note that the matrix must be square, that is, with the same number of rows and columns.

• Cofactor: It is the value obtained by multiplying an element of the matrix by the determinant of the remaining square submatrix. The sign of the cofactor depends on the position of the element in the matrix.

Examples and Cases

• Example of 1x1 Determinant: Consider the matrix A = [3]. The determinant of this matrix is |A| = 3, since it has only one element.

• Using 1x1 Determinant for Calculation Verification: The determinant of a 1x1 identity matrix A is always equal to 1. This property can be useful, for example, to check if the determinant calculation is correct, simply by calculating the determinant of the corresponding identity matrix of the same order.

• Relation between 1x1 Determinant and Scalar Multiplication: The determinant of a 1x1 matrix A = [k] (where k is a constant) is equal to k, that is, |A| = k. This highlights the property of determinants of scalar matrices.

Detailed Summary

Relevant Points

• Definition of 1x1 Determinant: This is an essential concept that allows us to calculate determinants for matrices of higher order. The key to remember is that the determinant of a 1x1 matrix is the element of the matrix itself. Therefore, if the matrix is A = [a], |A| = a.

• Difference between Matrix and Determinant: Matrices are broader structures that contain individual elements. Understanding the difference between a matrix and its determinant is crucial. A matrix can have many elements, but its determinant is always a single number.

• Introduction to Determinant Calculation: Working with 1x1 determinants is an essential foundation for advancing to higher orders. This introductory step familiarizes us with the structure and properties of determinants, preparing us for more advanced topics later on.

Conclusions

• Determinants and Matrices are Linked Concepts: Understanding determinants and matrices is a requirement to deepen the study of linear algebra. Matrices are supported by determinants, and determinants are an essential function of matrices.

• 1x1 Determinant and Scalar Properties: The determinant of a 1x1 matrix is always the element of the matrix. This demonstrates the property of determinants of scalar matrices, where the determinant is equal to the element on the diagonal of the matrix.

• 1x1 Determinant, the Basis for Higher Order Determinants: The study of the 1x1 determinant provides the basis for understanding determinants of higher orders. A solid understanding of the calculation of 1x1 determinants is therefore imperative for the more advanced study of linear algebra.

Exercises

1. Calculation of 1x1 Determinant: Calculate the determinant of the matrix A = [5].

2. Relating 1x1 Determinant and 2nd Order Matrices: Consider the matrix B = [ [3, 4], [5, 6] ]. The matrix B can be decomposed into two 1x1 matrices: C = [3] and D = [6]. Verify if the determinant of B is equal to the product of the determinants of C and D.

3. Applications of Scalar Multiplication: If the matrix E = [ [2, -1], [3, 4] ] and F is a matrix obtained by multiplying matrix E by a scalar k, show that the determinant of F is equal to k² times the determinant of E. Use this result to calculate |F|, if |E| = 10 and k = 3.