Clarifications on Row Echelon Form and Reduced Row Echelon Form

Question

RREF -> Reduced Row Echelon Form

REF -> Row Echelon Form

So I'm kinda stuck here. I have a quiz coming up next Wednesday and I can't seem to fully understand Row operations and reductions. Here are some the things I need clarification on:

1. When I'm asked to reduce a system to RREF, do I HAVE to take it to REF first before breaking it down further to RREF or do I just go straight to RREF? I'm asking this because I kept getting wrong answers when I was solving an example and when I checked the solutions, they took it to REF first. So do I have to break it down to REF first? (Assuming the question doesn't need REF)

2. Is a system in REF if the number above or below the leading one isn't 0? E.g is this in REF? Why and Why not?

   1 5 7 8 9

   0 1 3 0 2

   0 0 1 9 0

I'm asking this because all the examples I've seen on REF has 0 below the leading one it and a random number above the leading one

Thank you.

Answer 1

Some definitions are in order. Please refer to the references or any linear programming text for more details.

Matrix diagonal entries are the entries $a_ij$ such that $i=j$. The matrix does not have to be square matrix.

The definitions below work even in the case of augmented matrix (just ignore the vertical line).

A matrix is Echelon Form if:

A - All zero rows (if any) have moved to the bottom.

B - The leading non-zero is farther to the right than the leading nonzero element in the row just above it.

C - In each column containing a leading nonzero element, the entries below that element the leading nonzero element are zero.

Note: The Row Echelon form is NOT unique.

A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:

A - Matrix is in row echelon form.

B - The leading entry in each nonzero row is a 1 (called a leading 1).

C - Each column containing a leading 1 has zeros everywhere else (above and below). Note: The Reduced Row Echelon form is unique.

The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination.

Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it.For a given matrix, despite the row echelon form not being unique, all row echelon forms and the reduced row echelon form have the same number of zero rows (if any) and the pivots are located in the same indices.

do I HAVE to take it to REF first before breaking it down further to RREF

It is natural that you do REF and proceed to RREF - See the examples. It does make sense and keep the work organized at least. In some cases, you could move immediately to RREF. I guess we can claim that every matrix in RREF is also an REF but the opposite is not always true.

Is a system in REF if the number above or below the leading one isn't 0? E.g is this in REF? Why and Why not?

In REF definition, there is no mention of the value of the entries above the leading 1s. The definition references the entries below the leading 1s only. In the example you have provided, the matrix satisfies the rules of REF only.

Example:

Some References:

1-Google Books-Linear Algebra: Theory and Applications

2-Wiki-https://en.wikipedia.org/wiki/Row_echelon_form See above notes for useful links.