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.
**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.