How to compute row-reduced echelon form and understand upper triangular matrices

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Discussion Overview

The discussion revolves around understanding the computation of row-reduced echelon form (RREF) and the concept of upper triangular matrices. Participants seek clarification on the definitions and properties of these mathematical structures, as well as the implications of elementary row operations.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in understanding how to compute the row-reduced echelon form.
  • Another participant explains that a matrix remains unchanged through elementary row operations, emphasizing the goal of achieving an upper triangular matrix.
  • Concerns are raised about a lack of examples in a referenced book, prompting requests for clarification on what constitutes an upper triangular matrix.
  • A definition of an upper triangular matrix is provided, stating that it contains zeros below the main diagonal.
  • There is a discussion about the relationship between changing a matrix and the corresponding system of equations, with some participants asserting that the matrix can change while adhering to elementary row operations.
  • A suggestion is made to write out a consistent system of equations and convert it to RREF to illustrate the operations involved.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concepts of row-reduced echelon form and upper triangular matrices. There is no consensus on the clarity of the definitions or the examples provided in the referenced book.

Contextual Notes

Some participants note the lack of examples in their textbooks, which may limit their understanding of the concepts discussed. There is also an acknowledgment of the need for clarity regarding the implications of elementary row operations on matrices and their corresponding systems of equations.

innightmare
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I am having problems with understanding the whole concept/how to compute the row-reduced echelon form.

Can someone please help me? Thanks
 
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A matrix remains unchanged after going through the elementary row operations, so the whole concept is to keep on multiplying rows and adding (or subtracting) them from other rows to give an upper triangular matrix.
 
The book that i have doesn't give examples nor is it clear about the upper triangular matrix. Can you PLEASE explain what's an upper triangular matrix?
 
It is just a matrix [tex]\{a_{ij}\}[/tex] where the terms for which [tex]i[/tex] is bigger than [tex]j[/tex] are all zero.
 
yes, but i thought you changed your matix after changing the equation pertaining to it
 
Yes it does, generally, but not if you change the system of equations in strict accordance with the elementary row operations. Chapter three of Wylie's and Barrett's Advanced Engineering Mathematics (sixth edition) has proofs, and most university libraries have that book, I think.
 
just write out a system of equations, any system which you know is consistent and solve it. now write out the matrix for it and get it into rrref form and you'll see that you're performing the same operation you've just taken out the xs
 
innightmare said:
The book that i have doesn't give examples nor is it clear about the upper triangular matrix. Can you PLEASE explain what's an upper triangular matrix?
An upper triangular matrix is a matrix that has only zeros below the "main diagonal".
In other words, the non-zero entries form a triangle and it is above the diagonal.
 

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