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Chapter 8: Problem 6

Find the determinant of the matrix. $$[-12]$$

Short Answer

Expert verified
The determinant of the matrix is -12.

Step by step solution

01

Identifying the Matrix

The given matrix is a 1x1 matrix, which is just a single number. Here, the matrix is \([-12]\).
02

Finding the Determinant

The determinant of a 1x1 matrix is simply the value of the element in the matrix. Therefore, the determinant of the given matrix is -12.

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Key Concepts

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

Understanding a 1x1 Matrix
A 1x1 matrix is the simplest type of matrix you can encounter. Unlike larger matrices that have rows and columns, a 1x1 matrix is just a single number enclosed in square brackets. For example:
  • Matrix: \([-12]\)
The simplicity of a 1x1 matrix means that many operations, such as finding the determinant, become straightforward and easy to grasp.
A 1x1 matrix is important as it often serves as a foundation for understanding larger matrices. Once you understand how to deal with a single number matrix, you can apply that knowledge to more complex matrices.
Calculating the Determinant of a Matrix
The determinant is a special number calculated from a square matrix. It provides useful information about the matrix, such as whether it is invertible or not.
For larger matrices, finding the determinant can involve a series of complex calculations. However, for a 1x1 matrix, the determinant is simply the number itself.
  • Given Matrix: \([-12]\)
  • Determinant: \(-12\)
Determinants have many applications, especially in solving systems of linear equations and in finding areas and volumes when dealing with geometric shapes.
The Role of Matrices in Linear Algebra
Linear algebra is a branch of mathematics focusing on linear equations, vectors, and matrices.
Matrices are used to represent and solve complex systems of equations, transforming these systems into simpler, more manageable forms.
In linear algebra:
  • Matrices store coefficients of linear equations.
  • Determinants help determine solutions' nature and existence.
Understanding matrices and their properties, including determinants, is crucial because they provide insights into many fields like physics, computer science, and engineering. Mastery of 1x1 matrices and their determinants lays a foundation for tackling more advanced topics in linear algebra.

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Most popular questions from this chapter

An augmented matrix that represents a system of linear equations (in the variables \(x\) and \(y\) or \(x, y,\) and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{llllr} 1 & 0 & 0 & \vdots & 3 \\ 0 & 1 & 0 & \vdots & -1 \\ 0 & 0 & 1 & \vdots & 0 \end{array}\right]$$

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(6,3),(10,3)$$

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} e^{-x} & x e^{-x} \\ -e^{-x} & (1-x) e^{-x} \end{array}\right|$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{rr} 7 x-3 y & +2 w=41 \\ -2 x+y & -w=-13 \\ 4 x+z-2 w & =12 \\ -x+y-x & =-8 \end{array}\right.$$
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