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Chapter 8: Problem 6

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

Short Answer

Expert verified
The determinant of the given matrix is -10.

Step by step solution

01

Identify the matrix

The given matrix is a 1x1 matrix as it only contains one number: \([-10]\)
02

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

PROPERTIES OF DETERMINANTS In Exercises 97-99, a property of determinants is given (\(A\) and \(B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by adding a multiple of a row of \(A\) to another row of \(A\) or by adding a multiple of a column of \(A\) to another column of \(A\), then \(|B| = |A|\). (a) \(\left| \begin{array}{r} 1 & -3 \\ 5 & 2 \end{array} \right| = \left| \begin{array}{r} 1 & -3 \\ 0 & 17 \end{array} \right|\) (b) \(\left| \begin{array}{r} 5 & 4 && 2 \\ 2 & -3 && 4 \\ 7 & 6 && 3 \end{array} \right| = \left| \begin{array}{r} 1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3 \end{array} \right|\)

In Exercises 17-20, use a graphing utility and Cramer's Ruleto solve (if possible) the system of equations. \(\begin{cases} 3x + 3y + 5z = 1 \\ 3x + 5y + 9z = 2 \\ 5x + 9y + 17z = 4 \end{cases}\)

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} 4x - y + z = -5 \\ 2x + 2y + 3z = 10 \\ 5x - 2y + 6z = 1 \end{cases}\)

A message written according to a secret code is called a ________.

In Exercises 71-76, evaluate the determinant(s) to verify the equation. \(\left| \begin{array}{r} w & x \\ cw & cx \end{array} \right| = 0\)
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