/*! 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 27 Suppose \(A\) is the \(3 \times ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 27

Suppose \(A\) is the \(3 \times 3\) zero matrix (with all zero entries). Describe the solution set of the equation \(A \mathbf{x}=\mathbf{0} .\)

Short Answer

Expert verified
The solution set is the entire space \(\mathbb{R}^3\), as any vector satisfies the zero matrix equation.

Step by step solution

01

Understand the Matrix Multiplication

The matrix \(A\) is a \(3 \times 3\) zero matrix, meaning that every element in it is zero. When this matrix is multiplied by any vector \(\mathbf{x}\) of size \(3 \times 1\), the result will be the zero vector of size \(3 \times 1\). This is due to the nature of zero matrices, where multiplying by any other matrix or vector results in a zero matrix/vector.
02

Set Up the Equation

The equation given is \(A \mathbf{x} = \mathbf{0}\). Since \(A\) is the zero matrix, we actually have \(\begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}\). Each row of the zero matrix results in an equation \(0 \cdot x_1 + 0 \cdot x_2 + 0 \cdot x_3 = 0\).
03

Analyze the Resulting System of Equations

The system of equations simplifies to \(0 = 0\), which is trivially true for any values of \(x_1, x_2, x_3\). This implies that the solution set is not constrained by \(A\).
04

Describe the Solution Set

Since the matrix \(A\) imposes no constraints on \(\mathbf{x}\), the solution set includes all possible vectors \(\mathbf{x}\in \mathbb{R}^3\). In other words, any vector in three-dimensional space is a solution, making the solution set infinite and equal to \(\mathbb{R}^3\).

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.

Zero Matrix
A zero matrix is a special type of matrix where every element is zero. It doesn鈥檛 matter what size the matrix is, as long as all values in it are zero, it鈥檚 called a zero matrix.

One of the key features of a zero matrix is its behavior in arithmetic operations, especially multiplication. Whenever a zero matrix is multiplied by another matrix or a vector, the result is always another zero matrix or zero vector.

This property is crucial when dealing with equations, as it has significant implications on the solution sets of those equations.
Matrix Multiplication
Matrix multiplication is an essential operation in linear algebra, allowing you to combine rows and columns of two matrices to produce a new matrix. To multiply a matrix by a vector, consider the rows of the matrix and the single column of the vector.

Take a matrix of appropriate dimensions (e.g., a 3x3 zero matrix) and multiply it with a compatible vector (e.g., a 3x1 vector), the process results in each element of the resultant vector being the sum of products of respective elements from the matrix row and vector column.

For example:
  • If you multiply a zero matrix by any vector, the products will all be zero, leading to a result of a zero vector.
  • This behavior simplifies many systems of equations and can help solve or simplify mathematical problems effortlessly.
System of Equations
A system of equations consists of multiple equations that you try to solve simultaneously. When a zero matrix is involved, the system becomes incredibly straightforward, as observed in this exercise.

The equation formed using a zero matrix, such as \(A\mathbf{x} = \mathbf{0}\), generates equations like \(0 = 0\).

This implies that the system imposes no actual constraints on the solution due to the multiplying nature of zero. Therefore, the solution is not bound to any specific values and represents all possible vectors in that space.

Such conditions describe an infinite solution set due to lack of restriction, indicating that any vector in three-dimensional space satisfies the equation \(A\mathbf{x} = \mathbf{0}\).

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

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\left[\begin{array}{l}{1} \\ {1} \\ {1}\end{array}\right]\) is a solution of \(A \mathbf{x}=\mathbf{0}\) .

In Exercises \(1-10\) , assume that \(T\) is a linear transformation. Find the standard matrix of \(T\) . \(T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}, \quad T\left(\mathbf{e}_{1}\right)=(1,3), \quad T\left(\mathbf{e}_{2}\right)=(4,-7), \quad\) and \(T\left(\mathbf{e}_{3}\right)=(-5,4),\) where \(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\) are the columns of the \(3 \times 3\) identity matrix.

Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\left[\begin{array}{r}{1} \\ {5} \\\ {-3}\end{array}\right],\left[\begin{array}{r}{-2} \\ {-9} \\\ {6}\end{array}\right],\left[\begin{array}{r}{3} \\ {h} \\\ {-9}\end{array}\right]\)

Balance the chemical equations in Exercises \(5-10\) using the vector equation approach discussed in this section. Alka-Seltzer contains sodium bicarbonate \(\left(\mathrm{NaHCO}_{3}\right)\) and citric acid \(\left(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\right) .\) When a tablet is dissolved in water, the following reaction produces sodium citrate, water, and carbon dioxide (gas): \(\mathrm{NaHCO}_{3}+\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} \rightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2}\)

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.