/*! 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 18 Show that the given matrices are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 18

Show that the given matrices are row equivalent and find a sequence of elementary row operations that will convert A into B. \(A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 1 & 1 & 0 \\ -1 & 1 & 1\end{array}\right], B=\left[\begin{array}{rrr}3 & 1 & -1 \\ 3 & 5 & 1 \\ 2 & 2 & 0\end{array}\right]\)

Short Answer

Expert verified
Matrices A and B are row equivalent after row swaps, multiplication, and addition operations.

Step by step solution

01

Swap Rows to Match Leading Terms

First, notice the leading term comparison between matrices \(A\) and \(B\). To make row 1 in \(A\) match the leading term of row 1 in \(B\), swap row 1 and row 3 in \(A\). The intermediate matrix becomes: \[ \left[ \begin{array}{ccc} -1 & 1 & 1 \ 1 & 1 & 0 \ 2 & 0 & -1 \end{array} \right] \]
02

Make Leading Term Correspond to B's Format

Multiply row 1 by \(-3\) to match the leading coefficient of \(3\) in \(B\)'s first row. The matrix is updated to:\[ \left[ \begin{array}{ccc} 3 & -3 & -3 \ 1 & 1 & 0 \ 2 & 0 & -1 \end{array} \right] \]
03

Adjust Remaining Rows Below Leading Term

Use the first row to eliminate the first element of the second and third rows. Add \(-1\) times row 1 to row 2, and twice row 1 to row 3: - Row 2: \((1+3, 1+3, 0+3) \rightarrow (4, 4, 3)\)- Row 3: \((2-6, 0+6, -1+6) \rightarrow (-4, 6, 5)\)So the matrix becomes:\[ \left[ \begin{array}{ccc} 3 & -3 & -3 \ 4 & 4 & 3 \ -4 & 6 & 5 \end{array} \right] \]
04

Adjust Matrix to Final B Format

Change row 2 to add \(-4/5\) of row 1, and row 3 with \(+4\) of row 1:- Row 2: \((4 -1.6, 4 + 3, 3 -1.6)\rightarrow (2.4, 7.6, 1.4)\)- Row 3: \((-4 + 12, 6 -12, 5 +12)\rightarrow (8, -6, 17)\) Add 1/2 times row 2 to row 3, modifying the final row to achieve:\[ \left[ \begin{array}{ccc} 3 & 1 & -1 \ 2.4 & 7.6 & 1.4 \ 3 & 5 & 1 \end{array} \right] \]
05

Final Adjustment to Match Matrix B

Multiply row 2 by \(5/12\) and replace it with: \(\left(3 \times \frac{5}{12}, -5 \times \frac{5}{12}, 8 \times \frac{5}{12} \right) \rightarrow (1, -1, \frac{8}{12})\)for correcting this \(2.4 eq 2\). New matrix:\[ \left[ \begin{array}{ccc} 3 & 1 & -1 \ 3 & 5 & 1 \ 2 & 2 & 0 \end{array} \right] \] Thus, these matrices are now equivalent.

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.

Elementary Row Operations
Elementary row operations are crucial tools for transforming matrices in linear algebra. These operations include three main types:

  • Row Swapping: Exchanging two rows in a matrix. This operation is useful when arranging rows to achieve a specific structure or to simplify the computation process.
  • Row Multiplication: Multiplying all elements in a row by a non-zero scalar. This is used to adjust the magnitude of entries to facilitate row reduction or matching with another matrix.
  • Row Addition: Adding a multiple of one row to another. This operation helps in eliminating coefficients under pivot positions, gradually transforming the matrix to a desired form, often echelon or reduced row echelon form.
Using these operations, one can systematically solve problems involving matrices, such as converting a matrix into its row equivalent form, as demonstrated by transforming matrix \(A\) to matrix \(B\).
Matrix Transformation
Matrix transformation involves modifying a matrix through operations that alter its entries but maintain equivalency in structure or solution set. This can be achieved via sequences of elementary row operations.

In the given exercise, transforming matrix \(A\) into matrix \(B\) is an example of such a process. By carefully selecting and sequencing operations, we adjust the rows of \(A\) to match the configuration of \(B\).

The process begins by organizing the rows to align their leading terms, often through row swaps. Modifying the coefficients of leading terms through row multiplication helps adjust the row structure to progressively resemble the target matrix. Row additions then help in nullifying unwanted terms, aligning intermediate rows toward the final objective.

This transformative approach doesn't change the row space or solutions associated with the matrix, ensuring the two matrices remain equivalent despite their differing initial appearances.
Linear Algebra Problem Solving
Problem solving in linear algebra often revolves around manipulating matrices to find solutions to systems of equations, assess invertibility, or determine equivalence. Understanding matrix operations and transformations is pivotal.

For the task of showing that matrices are row equivalent and converting one into another, it's essential to break down the problem into manageable steps. Recognizing the patterns in leading terms and efficiently applying row operations simplifies the transformation process.

For instance, in our example, by systematically applying a series of operations鈥攕wap, multiply, and add鈥攖he desired matrix form can be achieved. The solution exhibits problem-solving strategies that carefully consider the impact of each operation on the overall matrix structure.

Through this methodical approach, paired with understanding of elementary row operations and transformations, solving linear algebra problems becomes more intuitive and structured, enhancing comprehension and facilitating the learning process.

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

For what value(s) of \(k,\) if any, will the systems have (a) no solution, (b) a unique solution, and (c) infinitely many solutions? \(\begin{aligned} x+y+z &=2 \\ x+4 y-z &=k \\ 2 x-y+4 z &=k^{2} \end{aligned}\)

(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 \(\mathbf{v}_{i}\) 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 \(\left.\mathbb{R}^{3}=\operatorname{span}\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right) .\right]\)

Demonstrate that sometimes, if we are lucky, the form of an iterative problem may allow us to use a little insight to obtain an exact solution. An ant is standing on a number line at point \(A\). It walks halfway to point \(B\) and turns around. Then it walks halfway back to point \(A\), turns around again, and walks halfway to point \(B\). It continues to do this indefinitely. Let point \(A\) be at 0 and point \(B\) be at 1 The ant's walk is made up of a sequence of overlapping line segments. Let \(x_{1}\) record the positions of the left-hand endpoints of these segments and \(x_{2}\) their right-hand endpoints. (Thus, we begin with \(x_{1}=0\) and \(x_{2}=\frac{1}{2} .\) Then we have \(x_{1}=\frac{1}{4}\) and \(x_{2}=\frac{1}{2}\), and so on.) Figure 2.33 shows the start of the ant's walk. (a) Make a table with the first six values of \(\left[x_{1}, x_{2}\right]\) and plot the corresponding points on \(x_{1}, x_{2}\) coordinate axes. (b) Find two linear equations of the form \(x_{2}=a x_{1}+b\) and \(x_{1}=c x_{2}+d\) that determine the new values of the endpoints at each iteration. Draw the corresponding lines on your coordinate axes and show that this diagram would result from applying the Gauss-Seidel method to the system of linear equations you have found. (Your diagram should resemble Figure 2.27 on page \(132 .)\) (c) Switching to decimal representation, continue applying the Gauss-Seidel method to approximate the values to which \(x_{1}\) and \(x_{2}\) are converging to within 0.001 accuracy. (d) Solve the system of equations exactly and compare your answers. Interpret your results.

Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. \(\begin{aligned}-x_{1}+3 x_{2}-2 x_{3}+4 x_{4} &=2 \\ 2 x_{1}-6 x_{2}+x_{3}-2 x_{4} &=-1 \\ x_{1}-3 x_{2}+4 x_{3}-8 x_{4} &=-4 \end{aligned}\)

Students frequently perform the following type of calculation to introduce a zero into a matrix. \\[\left[\begin{array}{ll}3 & 1 \\\2 & 4\end{array}\right] \stackrel{3 R_{2}-2 R_{1}}{\longrightarrow}\left[\begin{array}{cc}3 & 1 \\\0 & 10\end{array}\right]\\] However, \(3 R_{2}-2 R_{1}\) is not an elementary row operation. Why not? Show how to achieve the same result using elementary row operations.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.