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

In summary: This concept is important in computing the row-reduced echelon form of a matrix, which involves multiplying and adding (or subtracting) rows to create an upper triangular matrix. The book the person is using does not provide clear examples or explanations of this concept, but they suggest looking at chapter three of Wylie's and Barrett's Advanced Engineering Mathematics (sixth edition) for proofs and more information. The other person suggests writing out a system of equations, solving them, and then putting the matrix into row-reduced echelon form to see how the operations are performed.
  • #1
innightmare
35
0
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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
yes, but i thought you changed your matix after changing the equation pertaining to it
 
  • #6
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.
 
  • #7
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
 
  • #8
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.
 

1. What is a row-reduced echelon form?

A row-reduced echelon form (RREF) is a specific form of a matrix where the leading coefficient (first non-zero number) of each row is a 1, and all other numbers in the same column are 0. This form is useful for solving systems of linear equations and performing other operations on matrices.

2. How is a matrix transformed into row-reduced echelon form?

To transform a matrix into RREF, we use elementary row operations such as multiplying a row by a non-zero number, swapping two rows, or adding a multiple of one row to another. These operations do not change the solution to the system of equations represented by the matrix.

3. What are the benefits of using row-reduced echelon form?

RREF is useful for solving systems of linear equations because it allows for easier identification of the number of solutions (one, infinite, or none) and the values of the variables. It also simplifies matrix operations and makes it easier to find the inverse of a matrix.

4. Can any matrix be transformed into row-reduced echelon form?

Not all matrices can be transformed into RREF. For example, matrices with all zero rows cannot be transformed into RREF. Additionally, some matrices may require more complex operations to reach RREF, while others may not have a unique RREF.

5. How is row-reduced echelon form used in other fields of science?

RREF is used in various fields of science, such as engineering, physics, and computer science, to solve systems of linear equations and perform matrix operations. It is also used in data analysis and machine learning to simplify and manipulate large datasets.

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