/*! 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 10 Use elementary row operations to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

Use elementary row operations to reduce the given matrix to ( a) row echelon form and ( \(b\) ) reduced row echelon form. \(\left[\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right]\)

Short Answer

Expert verified
Row echelon form: \(\begin{bmatrix} 1 & \frac{3}{4}\\ 0 & -\frac{1}{2} \end{bmatrix}\). Reduced row echelon form: \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\).

Step by step solution

01

Identify Leading Entry

Start with the given matrix \(\begin{bmatrix}4 & 3 \ 2 & 1 \end{bmatrix}\). The leading entry in the first row is 4. We will first aim to make this leading entry a pivot (a 1), and later use it to eliminate the leading entry below it.
02

Create Pivot in Row 1

Divide the entire first row by 4 to create a pivot of 1. The new first row becomes \(\left[1, \frac{3}{4}\right]\). The matrix now looks like \(\begin{bmatrix} 1 & \frac{3}{4} \ 2 & 1 \end{bmatrix}\).
03

Eliminate Below Pivot in Column 1

Utilize the pivot in row 1 to eliminate the leading entry (2) in row 2 by subtracting 2 times the new row 1 from row 2. Perform the operation: Row 2 = Row 2 - 2 * Row 1. This gives a new row 2 of \(\left[0, -\frac{1}{2}\right]\). The matrix now transforms to \(\begin{bmatrix} 1 & \frac{3}{4} \ 0 & -\frac{1}{2} \end{bmatrix}\).
04

Result in Row Echelon Form

The matrix \(\begin{bmatrix} 1 & \frac{3}{4} \ 0 & -\frac{1}{2} \end{bmatrix}\) is in row echelon form, with leading 1s and zeros below them.
05

Create Pivot in Row 2

In row 2, divide by \(-\frac{1}{2}\) to form a pivot. The second row becomes \([0, 1]\). The matrix then updates to \(\begin{bmatrix} 1 & \frac{3}{4} \ 0 & 1 \end{bmatrix}\).
06

Eliminate Above Pivot in Column 2

Eliminate the \(\frac{3}{4}\) in row 1, column 2 by subtracting \(\frac{3}{4} \times\) Row 2 from Row 1. Perform the operation: Row 1 = Row 1 - \(\frac{3}{4} * \) Row 2. This changes row 1 to \([1, 0]\) and leaves the matrix as \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
07

Result in Reduced Row Echelon Form

The final matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\) is in reduced row echelon form with pivots as the only non-zero entries in rows.

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 fundamental to transforming matrices and solving linear equations. They allow us to manipulate matrices to achieve more desired forms without changing the underlying linear system. There are three essential types of these operations:
  • Row Switching: Interchanging two rows within a matrix. This operation helps in organizing the matrix for easier manipulation.
  • Row Multiplying: Multiplying all elements of a row by a non-zero scalar. It scales the entries, often used to create leading ones (pivots).
  • Row Addition: Adding a multiple of one row to another. This can eliminate specific entries and simplify the matrix.
These operations maintain the solutions of the system being solved. They are reversible, which ensures the integrity and validity of matrix transformations. Understanding and applying these operations is crucial when performing row reduction.
Row Echelon Form
Row echelon form is a type of matrix with specific characteristics that make solving linear systems easier. In this form, the matrix follows criteria:
  • All nonzero rows appear above any rows of all zeros.
  • The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
  • The leading entry in any nonzero row is 1, often referred to as a pivot.
Achieving row echelon form generally involves using elementary row operations. The objective is to create zeros below the pivots, simplifying the matrix structure and preparing it for further reduction or back substitution. This form is not unique for any given matrix but provides a consistent way to start solving systems.
Reduced Row Echelon Form
Reduced row echelon form is a further refined version of row echelon form that offers additional simplicity for solving linear systems. The criteria for a matrix in this form include:
  • Every leading entry of 1 is the only nonzero element in its column.
  • The leading 1s move to the right as you progress through the rows, similar to row echelon form.
  • All zero rows, if any, are located at the bottom of the matrix.
Achieving this form often involves returning to earlier rows to eliminate non-pivot entries, ensuring that every column containing a pivot is simplified entirely to that pivot. This form is unique for any matrix, unlike row echelon form, and provides a direct path to the solutions of a system of linear equations. It is highly valuable for understanding dependencies within the set of equations and confirming the number of unique solutions.

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

Apply Jacobis method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you like. $$\begin{aligned}20 x_{1}+x_{2}-x_{3} &=17 \\\x_{1}-10 x_{2}+x_{3} &=13 \\\\-x_{1}+x_{2}+10 x_{3} &=18\end{aligned}$$

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p^{*}}\). $$\begin{array}{l} x+2 y=1 \text { over } \mathbb{Z}_{3} \\ x+y=2 \end{array}$$

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)

Recall that the cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is a vector \(\mathbf{u} \times \mathbf{v}\) that is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). (See Exploration: The Cross Product in Chapter \(1 .\) ) If \\[ \mathbf{u}=\left[\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}\right] \text { and } \mathbf{v}=\left[\begin{array}{l} v_{1} \\ v_{2} \\ v_{3} \end{array}\right] \\] show that there are infinitely many vectors \\[ \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \\] that simultaneously satisfy \(\mathbf{u} \cdot \mathbf{x}=0\) and \(\mathbf{v} \cdot \mathbf{x}=0\) and that all are multiples of \\[ \mathbf{u} \times \mathbf{v}=\left[\begin{array}{l} u_{2} v_{3}-u_{3} v_{2} \\ u_{3} v_{1}-u_{1} v_{3} \\ u_{1} v_{2}-u_{2} v_{1} \end{array}\right] \\]

Give examples of homogeneous systems of \(m\) linear equations in \(n\) variables with \(m=n\) and with \(m>n\) that have (a) infinitely many solutions and (b) unique solution.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

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.