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I'm slightly confused by the definition of row echelon form. For example, is this matrix considered to be in row echelon form?

$$\begin{pmatrix} 1 & 2 & 5 & 8 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$

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This is a good question. Many people talk about row reduced echelon form, but it is not clear when exactly we can consider a matrix in RREF for all cases. –  Axl Jun 2 at 11:01
    
There's here also a good explanation: stattrek.com/matrix-algebra/echelon-form.aspx –  Axl Jun 2 at 11:08

1 Answer 1

Yes, that matrix is indeed in row echelon form. There are three conditions for row echelon form

  1. All zero rows are at the bottom of the matrix.
  2. All rows begin with a leading $1$.
  3. The leading entry of a row is strictly to the right to any leading entries from above rows.

Your matrix satisfies 1. because there are no zero rows. It satisfies 2. because all the entries begin with leading $1$s. It also satisfies 3. The leading entry of the second row is in position $3$ which is to the right of the leading entry in row $1$. The leading entry of row $3$ is also to the right of the leading ones above it.

Many matrices, especially if they are not square, will not have all the leading ones along the main diagonal, and that's perfectly O.K. As long as they satisfy the definition give above, they will be in row echelon form.

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