Identifying matrices as REF, RREF, or neither

In summary, identifying matrices as Row Echelon Form (REF), Reduced Row Echelon Form (RREF), or neither involves examining specific characteristics. A matrix is in REF if it has leading entries (pivots) that are all 1s, with each leading entry positioned to the right of the one in the row above, and all rows of zeros are at the bottom. A matrix is in RREF if it meets the REF criteria and additionally requires that each leading 1 is the only non-zero entry in its column. If a matrix fails to meet the criteria for either form, it is classified as neither.
  • #1
crememars
15
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Thread moved from the technical math forums to the schoolwork forum
TL;DR Summary: we are given a set of coefficient matrices (shown below) and we need to determine whether they are in REF, RREF, or neither.

Hello! I am having a lot of trouble identifying these matrices, and using the criteria checklist is not helping very much. Here is what I am working with:

Matrix A =
0 0 1
0 0 0
0 0 0

*I think this would be RREF. It has a leading 1 with no non-zero entries above or below it. The two zero rows are confusing me a little though.

Matrix B =
0 0
0 0
0 0

*This one has no leading entries at all, so does it automatically classify as neither REF nor RREF?

Matrix C =
0 0 1

*This matrix has only one row. We did not learn much about exceptions in class, but I feel as if matrices consist of at least more than two equations. Therefore, this matrix should be in neither form. If my reasoning is wrong, then I think that this might be RREF, since there is a leading 1 with no non-zero entries below or above it.

I would sincerely appreciate any help with these problems. Thank you!
 
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  • #2
crememars said:
*This matrix has only one row. We did not learn much about exceptions in class, but I feel as if matrices consist of at least more than two equations.
A matrix can have as few as one row or as few as one column. A matrix can represent a system of one or more equations, but it does not consist of equations.

How does your book define the terms REF (row-echelon form) and RREF (reduced row-echelon form)?
Since all three matrices you showed can't be simplified further, I would say that all three are RREF.
 
  • Informative
Likes berkeman

FAQ: Identifying matrices as REF, RREF, or neither

What is the difference between REF and RREF?

REF (Row Echelon Form) and RREF (Reduced Row Echelon Form) are both forms of matrices used to simplify linear systems. In REF, each leading entry of a row is to the right of the leading entry of the row above it, and all entries below a leading entry are zero. In RREF, in addition to the properties of REF, each leading entry is 1 and is the only nonzero entry in its column.

How can I determine if a matrix is in REF?

A matrix is in REF if it satisfies the following conditions: 1) All nonzero rows are above any rows of all zeros, 2) The leading entry of each nonzero row is strictly to the right of the leading entry of the row above it, and 3) All entries below a leading entry are zeros.

What additional criteria must a matrix meet to be in RREF?

In addition to the criteria for REF, a matrix must meet the following conditions to be in RREF: 1) The leading entry in each nonzero row must be 1, and 2) Each leading 1 must be the only nonzero entry in its column.

Can a matrix be neither REF nor RREF?

Yes, a matrix can be neither REF nor RREF if it does not satisfy the criteria for either form. For example, if the leading entries are not strictly to the right of the leading entries in the rows above, or if there are nonzero entries below a leading entry, the matrix is neither in REF nor in RREF.

What steps can I take to convert a matrix to REF or RREF?

To convert a matrix to REF, use row operations to ensure that all nonzero rows are above any rows of all zeros, and that each leading entry is to the right of the leading entry in the row above. To further convert to RREF, make sure each leading entry is 1 and that it is the only nonzero entry in its column. The row operations include row swapping, multiplying a row by a nonzero scalar, and adding or subtracting multiples of rows from each other.

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