Example of (echelon matrix/matrix in (row) echelon form)

Examples of matrices in row echelon form:

1. $$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
   — Pivot positions: (1,1) and (2,3). Zero row at bottom.

2. $$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix}$$
   — Already in echelon form (also upper triangular).

3. $$\begin{pmatrix} 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
   — First column has no pivot; pivots in columns 2 and 4.

Non-examples:

• $$\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$$ — entry below pivot is not zero
• $$\begin{pmatrix} 0 & 0 \\ 1 & 2 \end{pmatrix}$$ — zero row is above a nonzero row