/*! 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 20 What is the net effect of perfor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

What is the net effect of performing the following sequence of elementary row operations on a matrix (with at least two rows)?$$R_{2}+R_{1}, R_{1}-R_{2}, R_{2}+R_{1},-R_{1}$$

Short Answer

Expert verified
The net effect is swapping row 1 and row 2.

Step by step solution

01

Understanding the Problem

We need to determine the net effect of the given sequence of row operations on a matrix with at least two rows. The operations are: (1) add row 1 to row 2, (2) subtract row 2 from row 1, (3) add row 1 to row 2, and finally, (4) negate row 1.
02

Apply First Operation

The first operation is to add row 1 to row 2. This changes row 2 to the sum of row 1 and row 2. In symbols, if the original row 2 is represented as \(R_2\), it becomes \(R_2 + R_1\).
03

Apply Second Operation

The second operation is to subtract the modified row 2 from row 1. The initial row 1 is now modified to \(R_1 - (R_2 + R_1) = -R_2\).
04

Apply Third Operation

The third operation adds the modified row 1 to the new row 2. Thus, row 2 becomes \((R_2 + R_1) + (-R_2) = R_1\).
05

Apply Fourth Operation

The final operation is to negate the modified row 1. After the second operation, row 1 was \(-R_2\), so negating it gives \(-(-R_2) = R_2\).
06

Determine the Net Effect

After applying all operations, row 1 becomes \(R_2\) and row 2 becomes \(R_1\). Therefore, the net effect of these operations is to swap row 1 and row 2.

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.

matrix manipulation
Matrix manipulation involves altering matrices using various operations to achieve a specific goal. It is crucial when solving linear algebra problems, especially those involving systems of equations.
  • When manipulating matrices, you apply operations like addition, subtraction, and multiplication directly to the rows or columns.
  • Elementary row operations are a fundamental part of this process. These operations help transform matrices into easier forms, such as row-echelon or reduced row-echelon forms.
  • Effective manipulation of matrices can simplify complex systems, making it easier to solve for unknown variables.
A solid understanding of matrix manipulation enables you to handle more advanced tasks like finding matrix inverses or determinants. Familiarity with these operations can greatly enhance your problem-solving toolkit in linear algebra.
row operations sequence
The sequence of row operations is the order in which we apply these manipulations to a matrix. Each step in the sequence can alter the outcome significantly.
  • In our exercise, a specific order of operations is applied to a two-row matrix.
  • The operations were: adding row 1 to row 2, subtracting the resulting row 2 from row 1, adding row 1 to row 2 again, and negating row 1.
  • This sequence effectively transformed the matrix by changing and swapping the content of the rows.
Careful attention is necessary when following a row operations sequence because altering the order can lead to different results. Mastery of this concept is highly beneficial for tasks like row-reducing matrices to find solutions to matrix equations.
swapping rows in matrices
Swapping rows is an essential operation in matrix manipulation that involves exchanging the positions of two entire rows within a matrix. This can alter the structure without changing the determinant of the matrix.
  • In our original exercise, a sequence of operations resulted in the final outcome of swapping row 1 with row 2.
  • Swapping rows is commonly used when aiming to simplify a matrix, aiding in tasks such as solving linear equations or inverting matrices.
  • Despite the physical swap, the values in the rows remain unchanged, preserving certain matrix properties.
Understanding how and when to swap rows effectively can be a powerful skill, especially in simplifying matrix forms for easier computation and solving.

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

Let \(\mathbf{p}=\) \(\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \mathbf{q}=\left[\begin{array}{r}0 \\ 1 \\ -1\end{array}\right], \mathbf{u}=\left[\begin{array}{r}2 \\ -3 \\ 1\end{array}\right],\) and \(\mathbf{v}=\left[\begin{array}{r}0 \\ 6 \\ -1\end{array}\right]\) Show that the lines \(x=p+s u\) and \(x=q+t v\) are skew lines. Find vector equations of a pair of parallel planes, one containing each line.

Set up and solve an appropriate system of linear equations to answer the questions. From elementary geometry we know that there is a unique straight line through any two points in a plane. Less well known is the fact that there is a unique parabola through any three noncollinear points in a plane. For each set of points below, find a parabola with an equation of the form \(y=a x^{2}+b x+c\) that passes through the given points. (Sketch the resulting parabola to check the validity of your answer.) (a) \((0,1),(-1,4),\) and (2,1) (b) \((-3,1),(-2,2),\) and (-1,5)

A coffee merchant sells three blends of coffee. A bag of the house blend contains 300 grams of Colombian beans and 200 grams of French roast beans. A bag of the special blend contains 200 grams of Colombian beans, 200 grams of Kenyan beans, and 100 grams of French roast beans. A bag of the gourmet blend contains 100 grams of Colombian beans, 200 grams of Kenyan beans, and 200 grams of French roast beans. The merchant has on hand 30 kilograms of Colombian beans, 15 kilograms of Kenyan beans, and 25 kilograms of French roast beans. If he wishes to use up all of the beans, how many bags of each type of blend can be made?

Determine whether the lines \(x=\) \(\mathbf{p}+\) su and \(\mathbf{x}=\mathbf{q}+\) tv intersect and, if they do, find their point of intersection. $$\mathbf{p}=\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right], \mathbf{q}=\left[\begin{array}{r} -1 \\ 1 \\ -1 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 2 \\ 3 \\ 1 \end{array}\right]$$

(a) Suppose that vector \(\mathbf{w}\) is a linear combination of vectors \(\mathbf{u}_{1}, \ldots, \mathbf{u}_{k}\) and that each \(\mathbf{u}_{i}\) is a linear combination of vectors \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{m^{*}}\) Prove that \(\mathbf{w}\) is a linear combination of \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{m}\) and therefore \(\operatorname{span}\left(\mathbf{u}_{1}, \ldots, \mathbf{u}_{k}\right) \subseteq \operatorname{span}\left(\mathbf{v}_{1}, \ldots, \mathbf{v}_{m}\right)\) (b) In part (a), suppose in addition that each \(\mathbf{v}_{j}\) is also a linear combination of \(\mathbf{u}_{1}, \ldots, \mathbf{u}_{k} .\) Prove that \(\operatorname{span}\left(\mathbf{u}_{1}, \ldots, \mathbf{u}_{k}\right)=\operatorname{span}\left(\mathbf{v}_{1}, \ldots, \mathbf{v}_{m}\right)\) (c) Use the result of part (b) to prove that$$\mathbb{R}^{3}=\operatorname{span}\left(\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\right)$$ [Hint: We know that \(\mathbb{R}^{3}=\operatorname{span}\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right) .\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.