Echelon Form Clarification

Question

I know that a $3\times 3$ has 7 echelon forms, I found them all out but I was wondering why this is not an echelon form $$ \begin{pmatrix} 0 &0 &1 \\ 0& 0 &0 \\ 0&0 & 0 \end{pmatrix} $$

Answer 1

It is in row echelon form.

Row echelon form of rank $1$ are of the form of

$$ \begin{pmatrix} 0 &0 &1 \\ 0& 0 &0 \\ 0&0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 & * \\ 0& 0 &0 \\ 0&0 & 0 \end{pmatrix}, \begin{pmatrix} 1 &* &* \\ 0& 0 &0 \\ 0&0 & 0 \end{pmatrix} $$

It satisfy the following criteria

• all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix), and
• the leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it (some texts add the condition that the leading coefficient must be $1$.

There are $8$ of them rather than $7$ of them.