Let A be the matrix: [1 2 0 3] [0 0 0 0] [0 0 1 2] Is A in RREF?

[AsymptoticCoder]

I've taken two upper division courses in linear algebra, and yet a trivial problem has arisen that I am having trouble answering! haha

Let A be the matrix:

[1 2 0 3]
[0 0 0 0]
[0 0 1 2]

Is A in RREF? I say yes, however, the majority of students whose papers I am grading say no. If they are correct, what is the reason?

TIA,
Ryan

 

[matt grime]

the row of zeroes should be on the bottom.

 

[M98Ranger]

Yes, A is in RREF (reduced row echelon form). To be in RREF, a matrix must satisfy three conditions:

1. All leading entries (the first non-zero entry in each row) must be 1.
2. All entries above and below a leading entry must be 0.
3. The leading entry in each row must be to the right of the leading entry in the row above it.

In the given matrix A, the first row has a leading entry of 1, the second row has all entries equal to 0, and the third row has a leading entry of 1 to the right of the leading entry in the row above it. Therefore, A satisfies all three conditions and is in RREF.

It is possible that the majority of students are confused because the second row has all entries equal to 0, which may seem unusual. However, this does not violate any of the conditions for RREF. In fact, it is common for matrices to have all zero rows in RREF.

Overall, it is important to carefully check all three conditions to determine if a matrix is in RREF. It is also helpful to practice identifying matrices in RREF to become more familiar with the concept.