Clarification on what is considered reduced row-echelon form

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In summary, the conversation is about the definition of RREF (Row Echelon Form) in linear algebra and whether two given matrices, (0 0, 0 0) and (0 1, 0 0), are considered RREF. The definition of RREF includes having nonzero rows above rows of all zeroes and the leading coefficient always being to the right of the leading coefficient of the row above it. The third condition for reduced row echelon form is that every leading coefficient is 1 and is the only nonzero entry in its column. The second matrix, (0 1, 0 0), appears to satisfy all three conditions, but there is confusion about whether a matrix with all zeroes can still be
  • #1
maximade
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Homework Statement


In a 2x2 matrix are these considered RREF?
(0 0, 0 0) and (0 1, 0 0)
 
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  • #2
Well, is the definition of RREF?
 
  • #3
According to wikipedia, it is:
In linear algebra a matrix is in row echelon form if

* All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes, and
* The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
 
  • #4
Are you unsure whether these two matrices have those properties then?

And what is the third condition, for reduced row echelon form?
 
  • #5
Every leading coefficient is 1 and is the only nonzero entry in its column.

From the looks of it, it looks like it follows those properties, but I'm just confused if it is still rref if everything is 0, since it doesn't have a leading 1.
 
  • #6
The definition doesn't say that there must be nonzero rows (for your first matrix). In your second matrix
[0 1]
[0 0]
The leading entry in the first row is a 1.
 

1. What is reduced row-echelon form?

Reduced row-echelon form is a specific way of organizing a matrix in linear algebra. It is achieved by performing a series of row operations (such as swapping rows, multiplying a row by a non-zero constant, and adding one row to another) on a matrix until it meets certain criteria. This form is useful in solving systems of linear equations and finding the rank and nullity of a matrix.

2. How is reduced row-echelon form different from row-echelon form?

Reduced row-echelon form and row-echelon form are two different ways of organizing a matrix. Row-echelon form is achieved by performing the same row operations as reduced row-echelon form, but it does not have the additional criteria of having leading ones in every column. In other words, reduced row-echelon form is a stricter version of row-echelon form.

3. What are the criteria for a matrix to be in reduced row-echelon form?

For a matrix to be in reduced row-echelon form, it must meet three criteria: 1) All leading coefficients (the first non-zero entry in each row) must be equal to 1, 2) All entries above and below leading coefficients must be equal to 0, and 3) Each leading coefficient must be the only non-zero entry in its column.

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

Yes, any matrix can be transformed into reduced row-echelon form by performing a series of row operations. However, the resulting matrix may not be unique, as there may be multiple ways to achieve reduced row-echelon form for a given matrix.

5. What is the purpose of using reduced row-echelon form?

Reduced row-echelon form is useful in solving systems of linear equations, as it simplifies the process by reducing the matrix into a form where the solution can be easily read off. It also helps in determining the rank and nullity of a matrix, which are important concepts in linear algebra.

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