Problem 6 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \([-10]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is -10.

Step 1: Identify the matrix

The given matrix is a 1x1 matrix as it only contains one number: \([-10]\)

Step 2: Calculate the determinant of the matrix

The determinant of a 1x1 matrix A is simply the number in it. So the determinant of the given matrix is -10

Key Concepts

Understanding 1x1 Matrices

A 1x1 matrix is the simplest type of matrix you can encounter in mathematics. It contains only one element, making it a single-number matrix. Thus, instead of seeing multiple rows and columns, you have just one cell containing a single value. In this exercise, the matrix given is

• \([-10]\)

This is a typical 1x1 matrix.
Because of its simplicity, working with a 1x1 matrix often serves as an excellent introduction to the concepts of matrices, determinants, and other matrix-related operations.

How to Calculate the Determinant of 1x1 Matrices

Calculating the determinant of a matrix generally involves more complex processes, especially as the matrix gets larger with more rows and columns. However, with a 1x1 matrix, this process is extremely straightforward.
For a 1x1 matrix, the determinant is simply the value of the single element present.
So, for the matrix

• \([-10]\)

The determinant is

• -10

Since there's no need for any complex steps or operations, the process of determining the determinant of a 1x1 matrix showcases how basic the initial stages of linear algebra can be.

The Role of Matrices in Mathematics

Matrices are not only a mathematical concept but a fundamental tool for various applications in science and engineering. They provide a framework for representing and solving complex problems, ranging from computer graphics to solutions of simultaneous equations.
Here are some roles matrices play in mathematics:

• "Representations of Linear Transformations": Many operations in graphics and physics can be described through matrices.
• "Solution of Linear Systems": Matrices are often used to find solutions to systems of linear equations.
• "Data Organization": Storing information in a structured format allows for easier manipulation and retrieval.

Thus, while learning about basic matrices, such as the 1x1, it's essential to appreciate their broader implications and usefulness across different fields.