/*! 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 6 Determine whether the given matr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 6

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. \(\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\)

Short Answer

Expert verified
The matrix is in reduced row echelon form.

Step by step solution

01

Understand Row Echelon Form Criteria

A matrix is in row echelon form if the following conditions are met: - All nonzero rows are above any rows of all zeros. - The leading coefficient (first non-zero number from the left, also called a pivot) of a non-zero row is to the right of the leading coefficient of the row above it. - The leading coefficient is always 1, and all elements below it are zeros.
02

Analyze the Given Matrix for Row Echelon Form

Examine the matrix:\[\begin{array}{ccc}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 \\end{array}\]- The first leading coefficient appears in the third row at column 1.- The second leading coefficient appears in row 2 at column 2 and is to the right of the first leading coefficient.- The third leading coefficient appears in row 1 at column 3 and is to the right of the second leading coefficient.- All nonzero rows are above any zero rows.Since all conditions are met, the matrix is in row echelon form.
03

Check for Reduced Row Echelon Form

A matrix is in reduced row echelon form if:- It is in row echelon form.- All leading coefficients are 1.- Each leading coefficient is the only non-zero entry in its column.Inspect the matrix:\[\begin{array}{ccc}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 \\end{array}\]- All leading coefficients are 1.- Each leading coefficient is indeed the only non-zero entry in its column.Hence, the matrix is in reduced row echelon form.

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.

Reduced Row Echelon Form
Understanding the concept of the Reduced Row Echelon Form (RREF) is crucial when analyzing matrices. This form has additional criteria compared to the Row Echelon Form (REF). Not only does it have all the conditions of REF, but it also requires that:
  • Each leading coefficient is exactly 1.
  • Each leading coefficient is the only non-zero entry in its column.
When examining the matrix: \[\begin{bmatrix}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 \\end{bmatrix}\]we can see it fulfills the extra requirements. All leading coefficients are 1 and they stand alone, without any other non-zero numbers in their columns. This purity in column ensures there's no ambiguity about the independent nature of the equations represented by this matrix. So, ultimately, this matrix is indeed in RREF.
Leading Coefficient
The leading coefficient is fundamental to understanding matrices in Row Echelon Form or Reduced Row Echelon Form. Simply put, a leading coefficient, or pivot, is the first non-zero element in a row, reading from left to right. In our matrix,\[\begin{bmatrix}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 \\end{bmatrix}\]the leading coefficients are positioned at columns 3, 2, and 1 in descending row order.The properties of leading coefficients include:
  • Leading coefficients must be to the right of any leading coefficients in the rows above them.
  • In matrices that are in reduced row echelon form, leading coefficients must also be 1.
  • All elements below a leading coefficient are zeros.
These rules ensure that our matrix not only meets the criteria for REF but also smoothly transitions into RREF if each leading coefficient stands alone in its column.
Matrix Analysis
Matrix analysis is a powerful tool in understanding systems of equations, transformations, and more. When analyzing a matrix, we identify several key aspects:
  • **Zero and non-zero rows:** All non-zero rows must appear before any zero rows to qualify for row echelon structures.
  • **Order of leading coefficients:** Alignment of leading coefficients in strict left to right order from top to bottom is pivotal in identifying pivotal rows that can impact the overall solution.
  • **Column integrity:** In reduced echelon, each column that houses a leading coefficient should make it the only non-zero entry.
By analyzing the given matrix,\[\begin{bmatrix}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0 \\end{bmatrix}\]we see a carefully structured form that not only adheres to the row echelon form but also progresses to reduced echelon due to its cleanliness in leading coefficient placement and columnar zero entries beneath them. This matrix analysis, therefore, confirms its optimal structure for solving linear equations.

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

Balance tire chemical equation for each reaction. \(\mathrm{C}_{4} \mathrm{H}_{10}+\mathrm{O}_{2} \longrightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}\) (This reaction occurs when butane, \(C_{4} H_{10}\), burns in the presence of oxygen to form carbon dioxide and water.

Apply Jacobi's 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. $$\begin{aligned} 2 x_{1}+x_{2} &=5 \\ x_{1}-x_{2} &=1 \end{aligned}$$

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)

Apply Jacobi's 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. $$\begin{aligned} 7 x_{1}-x_{2} &=6 \\ x_{1}-5 x_{2} &=-4 \end{aligned}$$

Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. \(\begin{aligned}w+x+2 y+z & \qquad=1 \\\w-x-y+z &\qquad=0 \\\x+y &\qquad=-1 \\\w+x &\quad+z=2\end{aligned}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.