/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 111 Determine whether each of the fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 111

Determine whether each of the following statements is true or false: When a system of linear equations is represented by a square augmented matrix, the system of equations always has a unique solution.

Short Answer

Expert verified
The statement is false.

Step by step solution

01

Understanding the Problem

We need to determine whether a system of linear equations represented by a square augmented matrix always has a unique solution.
02

Defining Terms: Augmented Matrix and Square Matrix

A square matrix is a matrix with the same number of rows and columns. An augmented matrix includes the coefficients and constants from a system of equations.
03

Considering the Possibilities for Solutions

A system of linear equations can have one unique solution, infinitely many solutions, or no solution. A square augmented matrix does not guarantee a unique solution.
04

Applying Determinants and Rank

For a system of equations with a square matrix to have a unique solution, the determinant of the coefficient matrix should be non-zero. If the determinant is zero, the matrix is singular, and the system might be inconsistent (no solution) or have infinitely many solutions.
05

Analyzing Implications

Because the determinant could be zero, and the rank of the matrix may lead to inconsistencies, a square augmented matrix does not always imply a unique solution.
06

Concluding the Statement's Truth Value

Given the above analysis, the statement that a square augmented matrix always has a unique solution is false.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Augmented Matrix
In the world of linear algebra, an augmented matrix is a helpful tool when solving systems of linear equations. It is formed by appending the constants from the equations to the right side of the coefficient matrix. This transformation simplifies the process of solving the equations.
Augmented matrices are especially useful in methods such as Gaussian elimination. The primary aim is to use row operations, change the matrix's form, and make it easier to find solutions.
  • An augmented matrix can help efficiently represent a system of equations.
  • It contains all coefficients and constant terms, which makes it comprehensive and adaptable for computational methods.
  • The layout is crucial; it allows easy manipulation and recognition of the system's nature.
Understanding how to set up and utilize an augmented matrix is a fundamental skill in linear algebra, as this method is commonly used to determine solutions to the equations.
Square Matrix
A square matrix is a matrix with an equal number of rows and columns. This specific structure is especially significant because it is required to calculate certain mathematical properties, such as the determinant.
Square matrices are central to many discussions in linear algebra since they align perfectly with many operations, facilitating certain calculations. Here are some important details about square matrices:
  • They are essential when discussing the properties of linear transformations and are used extensively in operations.
  • A square matrix can represent linear transformations that map vector spaces onto themselves.
  • They are integral in solving linear equations, especially when determining if a unique solution exists.
Overall, square matrices simplify numerous processes within mathematics, making them a pivotal element of various calculations.
Determinant
The determinant is a numerical value that can be calculated from a square matrix. It is an important concept as it helps determine certain properties of the matrix, such as whether it is singular or invertible.
If a square matrix has a non-zero determinant, it is considered invertible, or non-singular, implying the system of equations has a unique solution. Here are key points about determinants:
  • A non-zero determinant indicates a unique solution exists for the system of equations.
  • A zero determinant means the matrix is singular, which could result in no solution or infinitely many solutions.
  • Determinants help in understanding the matrix's behavior and its potential solutions.
Calculating the determinant is a crucial step in analyzing square matrices and understanding whether unique solutions exist for the associated systems of equations.
Unique Solution
In the context of systems of linear equations, a unique solution means that there is exactly one set of values that satisfy all the equations simultaneously. Finding a unique solution is often the aim when solving systems, as it provides clear and definitive answers.
Unique solutions occur under specific conditions:
  • The coefficient matrix must be square, and its determinant must not be zero.
  • This implies the matrix is invertible, ensuring that a single solution is possible.
  • Recognizing when a system has a unique solution is crucial for effective problem-solving in linear algebra.
Understanding the conditions under which unique solutions exist helps in applying suitable techniques to solve systems of equations effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In calculus, the first steps when solving the problem of finding the area enclosed by a set of curves are similar to those for finding the feasible region in a linear programming problem. In Exercises \(95-98,\) graph the system of inequalities and identify the vertices, that is, the points of intersection of the given curves. $$\begin{aligned} &y \leq x+2\\\ &y \geq x^{2} \end{aligned}$$

Gary and Ginger decide to place \(10,000\) of their savings into investments. They put some in a money market account earning \(3 \%\) interest, some in a mutual fund that has been averaging \(7 \%\) a year, and some in a stock that rose \(10 \%\) last year. If they put \(3,000\) more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year, they will carn \(\$ 540\) in a year. How much money did they put in each of the three investments?

Ann would like to exercise one hour per day to burn calories and lose weight. She would like to engage in three activities: walking, step-up exercise, and weight training. She knows she can burn 85 calories walking at a certain pacc in 15 minutes, 45 calories doing the step-up exercise in 10 minutes, and 137 calories by weight training for 20 minutes. a. Determine the number of calories per minute she can burn doing each activity. b. Suppose she has time to exercise for only one hour ( 60 minutes). She sets a goal of burning 358 calories in one hour and would

Solve the system of equations using an augmented matrix. $$ \begin{aligned} x+3 y+2 z &=4 \\ 3 x+10 y+9 z &=17 \\ 2 x+7 y+7 z &=17 \end{aligned} $$ Solution: $$ \begin{array}{c} \text { Step 1: Write the system as an } \\ \text { augmented matrix. } \end{array}\left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \\ 3 & 10 & 9 & 17 \\ 2 & 7 & 7 & 17 \end{array}\right] $$ \(\begin{aligned} \text { Step 2: Reduce the matrix using } &\left[\begin{array}{lll|r}1 & 0 & -7 & -11 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 4\end{array}\right] \\ \text { Gaussian elimination. } & 0 \end{aligned}\) Step 3: Identify the answer: \(\quad x=7 t-11\) Infinitely many solutions. \(\quad y=-3 t+5\) \(z=t\) This is incorrect. What mistake was made?

In order for \(A_{m \times n}^{2}\) to be defined, what condition (with respect to \(m\) and \(n\) ) must be met?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.