**Definition:** The **(row) echelon form** of a matrix $A$ is any matrix $U$ in row echelon form such that $U$ is **row-equivalent** to $A$ (i.e., $U$ can be obtained from $A$ by a sequence of elementary row operations).

**Key points:**
- A matrix can have **many** echelon forms (depending on the sequence of row operations chosen).
- However, every matrix has exactly **one** reduced row echelon form (RREF).
- The **pivot positions** are the same in all echelon forms of a given matrix.
- The echelon form reveals: rank (number of pivots), consistency of the associated linear system, and which variables are free vs. basic.