Reduced row echelon form question

In summary, reduced row echelon form is a specific form in which rows with nonzero entries must have a leading 1, columns with leading 1s must have all other entries as zero, and each row above a row with a leading 1 must also have a leading 1 further to the left. The term "leading 1" refers to the first nonzero entry in a row or column. This form is also known as the "row echelon form".
  • #1
wumple
60
0
My book gives the following definition for reduced row echelon form:

1) If a row has nonzero entries, then the first nonzero entry is 1, called the leading 1 in this row.
2) If a column contains a leading 1, then all other entries in that column are zero.
3) If a row contains a leading 1, then each row above contains a leading 1 further to the left.

1 and 3 I understand, but 2 I don't fully understand - does that mean a 'column leading 1', as in if I start at the top of a column and go down the numbers then the first nonzero entry is a 1, or does it mean that if a column contains a 1 that is a leading 1 for its row?

Thanks!
 
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  • #2
No, it means a "row leading 1", as defined in 1). So, if a ROW has a leading 1, than in the COLUMN of that particular leading 1 all other entries are zero. Perhaps the wkikipedia definition is clearer.
 
  • #3
Thank you!
 

Related to Reduced row echelon form question

1. What is reduced row echelon form?

Reduced row echelon form (RREF) is a way of organizing a matrix in a specific way to make it easier to solve systems of linear equations. It is a method of performing Gaussian elimination, which involves using elementary row operations to transform a matrix into a simpler form.

2. Why is reduced row echelon form important?

RREF is important because it allows us to more easily solve systems of linear equations. It also provides a unique representation of a matrix and helps to identify the rank of a matrix, which is useful in many areas of mathematics and science.

3. How is reduced row echelon form different from row echelon form?

RREF is a more simplified form of row echelon form (REF). In RREF, each leading coefficient (the first non-zero entry) in a row is equal to 1, and each leading coefficient is the only non-zero entry in its column. In REF, the leading coefficients can be any non-zero number and there may be other non-zero entries in the same column.

4. What are the steps to convert a matrix into reduced row echelon form?

The steps to convert a matrix into RREF include: 1) Using elementary row operations to create zeros below each leading coefficient, 2) Using elementary row operations to create zeros above each leading coefficient, 3) Using elementary row operations to make each leading coefficient equal to 1, and 4) Using elementary row operations to create zeros in all other positions. This will result in the matrix being in reduced row echelon form.

5. Can any matrix be converted into reduced row echelon form?

Yes, any matrix can be converted into reduced row echelon form. However, some matrices may not have a unique RREF, and some may not have a RREF at all. This depends on the specific properties of the matrix, such as whether it is invertible or not.

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