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Chapter 8: Problem 5

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{llll|l} 1 & 0 & 4 & 3 & 0 \\ 0 & 1 & 3 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] $$

Short Answer

Expert verified
The matrix is in row echelon form only.

Step by step solution

01

Understand Matrix Forms

The two forms we are interested in are **row echelon form** and **reduced row echelon form**. In row echelon form, each nonzero row has more leading zeros than the previous row, the leading entry of each non-zero row (after the first) appears to the right of the leading entry of the previous row, and columns containing the leading coefficient of a row are called pivot columns. In the reduced row echelon form, in addition to the conditions for row echelon form, every leading coefficient is 1, and it is the only non-zero entry in its column.
02

Analyze Row Echelon Form

The matrix is in row echelon form if all non-zero rows are above any rows of all zeros and the leading entry of each non-zero row after the first occurs to the right of the leading entry of the previous row with clear pivot positions. The given matrix satisfies these conditions: the first leading entry is in column 1 and second row's leading entry is in column 2, and the last row is all zeros. Therefore, the matrix is in row echelon form.
03

Check for Reduced Row Echelon Form

Check if every leading coefficient is 1 (which they are in this case), and it is the only non-zero entry in its column. In the given matrix, columns 1 and 2 have leading ones, but they have other non-zero entries within those columns (e.g., column 3 and 4 for the first and second rows). Since these columns are not all zero except for the leading 1, the matrix is not in reduced row echelon form.
04

Conclusion

Since the matrix satisfies the conditions for row echelon form, but not for reduced row echelon form, it is identified as being in row echelon form only.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reduced Row Echelon Form
Reduced row echelon form (RREF) is a specific type of matrix form that makes solving equations efficient and systematic. To qualify as a reduced row echelon form, a matrix must meet several strict criteria:
  • Each leading entry in a row is 1.
  • Leading 1's are the only non-zero entries in their columns.
  • Each leading 1 is to the right of any leading 1s in the rows above.
  • Rows with all zero elements, if any, are at the bottom of the matrix.
These rules ensure that when performing operations on the matrix, each variable or unknown is isolated in a simple way. This simplification is essential in linear algebra for swiftly solving systems of linear equations. In practice, achieving RREF can be more demanding because it requires additional steps to transform a matrix from row echelon form into reduced row echelon form by clearing all entries above each leading 1.
Matrix Analysis
Matrix analysis is the study of a matrix's properties and its transformation using established mathematical procedures. In linear algebra, understanding a matrix's form provides vital clues to its characteristics and potential uses. Analysts assess things like:
  • Matrix dimensions, which capture how many rows and columns it has.
  • The arrangement of non-zero elements and patterns of zeros.
  • Special forms like row echelon form or reduced row echelon form.
Knowing and identifying these features allow mathematicians to decide the best strategy for problem-solving. Matrices in row echelon or reduced row echelon form simplify computations, making it easier to handle operations such as back substitution or the determination of rank and independence. Quickly interpreting these matrix traits can, therefore, save significant computational effort.
Pivot Positions
Pivot positions are the spots where the leading coefficients, typically 1s, are located in each row of a matrix in row echelon form. They are crucial because they help establish a matrix's structure and guide its transformations. Understanding pivot positions can help in:
  • Identifying independent leading variables.
  • Assessing a matrix's rank, as each pivot represents a column contributing to the row space.
  • Evaluating whether a matrix can be reduced further to a simpler form.
In a matrix like the one given, pivot positions are clear indicators of the steps needed to simplify or solve the matrix. Recognizing where these pivots lie informs how row operations can progress to reach row echelon or reduced row echelon form efficiently.
Leading Entry
The leading entry of a row in a matrix is the first non-zero element in that row. This concept is essential when organizing rows in either row echelon form or reduced row echelon form. Each leading entry serves a couple of important purposes:
  • It helps establish the structure of the matrix by denoting which columns have been resolved.
  • In row echelon matrices, it ensures that each leading entry appears to the right of the leading entry above it.
  • In reduced row echelon form, it should be the lone non-zero in its column.
Without a clear leading entry, it becomes much harder to systematically reduce a matrix or understand its behavior in response to linear transformations. The alignment and presence of leading entries indicate how near a matrix is to being fully simplified in its echelon forms.

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Most popular questions from this chapter

Consider the following definitions. A square matrix is said to be an upper triangular matrix if all of its entries below the main diagonal are zero and it is said to be a lower triangular matrix if all of its entries above the main diagonal are zero. For example,$$ E=\left[\begin{array}{rrr} 1 & 2 & 3 \\\ 0 & 4 & -9 \\ 0 & 0 & -5 \end{array}\right] $$ from Exercises 8 - 21 above is an upper triangular matrix whereas $$ F=\left[\begin{array}{ll} 1 & 0 \\\ 3 & 0 \end{array}\right] $$ is a lower triangular matrix. (Zeros are allowed on the main diagonal.) Discuss the following questions with your classmates. Given the matrix $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $$ write \(A\) as \(L U\) where \(L\) is a lower triangular matrix and \(U\) is an upper triangular matrix?

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{aligned} \sqrt{x+1}-y &=0 \\ x^{2}+4 y^{2} &=4 \end{aligned}\right. $$

In Exercises 7 - 18 , find the partial fraction decomposition of the following rational expressions. $$ \frac{-7 x^{2}-76 x-208}{x^{3}+18 x^{2}+108 x+216} $$

Consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. Show that \(S Y=X_{s}\) for any matrix \(Y\) of the form $$ Y=\left[\begin{array}{r} y \\ 150-y \end{array}\right] $$ This means that no matter how the distribution starts in Pedimaxus, if \(Q\) is applied often enough, we always end up with 100 people getting the Tribune and 50 people getting the Picayune.

Sketch the solution to each system of nonlinear inequalities in the plane. $$ \left\\{\begin{aligned} x^{2}-y^{2} & \leq 1 \\ x^{2}+4 y^{2} & \geq 4 \end{aligned}\right. $$
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