Row Echelon Form

48. Row Echelon Form

A matrix is in Row Echelon Form (REF) if it satisfies all of:

Pivot — each non-zero row has a leading non-zero entry called a pivot
Zeros below pivots — each pivot has zeros below it in its column
Rightward staircase — each pivot is strictly to the right of the pivot in the row above
Zero rows at bottom — any all-zero rows appear at the bottom

[\begin{matrix} 1 & 𝑎_{12} & 𝑎_{13} & 𝑎_{14} & 𝑏_{1} \\ 0 & 1 & 𝑎_{23} & 𝑎_{24} & 𝑏_{2} \\ 0 & 0 & 1 & 𝑎_{34} & 𝑏_{3} \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]

The three operations that don’t change the solution set of a system, used to drive a matrix toward REF:

These are applied via Gaussian elimination.

Example

System:

\begin{matrix} 2 𝑥 + 𝑦 + 𝑧 = 8 \\ - 3 𝑥 - 𝑦 + 2 𝑧 = - 11 \\ - 2 𝑥 + 𝑦 + 2 𝑧 = - 3 \end{matrix}

Augmented matrix:

[\begin{matrix} 2 & 1 & 1 & 8 \\ - 3 & - 1 & 2 & - 11 \\ - 2 & 1 & 2 & - 3 \end{matrix}]

Step 1: $𝑅_{1} \to \frac{1}{2} 𝑅_{1}$ :

[\begin{matrix} 1 & 0.5 & 0.5 & 4 \\ - 3 & - 1 & 2 & - 11 \\ - 2 & 1 & 2 & - 3 \end{matrix}]

Step 2: $𝑅_{2} \to 𝑅_{2} + 3 𝑅_{1}$ , $𝑅_{3} \to 𝑅_{3} + 2 𝑅_{1}$ :

[\begin{matrix} 1 & 0.5 & 0.5 & 4 \\ 0 & 0.5 & 3.5 & 1 \\ 0 & 2 & 3 & 5 \end{matrix}]

Step 3: $𝑅_{2} \to 2 𝑅_{2}$ :

[\begin{matrix} 1 & 0.5 & 0.5 & 4 \\ 0 & 1 & 7 & 2 \\ 0 & 2 & 3 & 5 \end{matrix}]

Step 4: $𝑅_{3} \to 𝑅_{3} - 2 𝑅_{2}$ :

[\begin{matrix} 1 & 0.5 & 0.5 & 4 \\ 0 & 1 & 7 & 2 \\ 0 & 0 & - 11 & 1 \end{matrix}]

Step 5: $𝑅_{3} \to - \frac{1}{11} 𝑅_{3}$ (now in REF):

[\begin{matrix} 1 & 0.5 & 0.5 & 4 \\ 0 & 1 & 7 & 2 \\ 0 & 0 & 1 & - \frac{1}{11} \end{matrix}]

Step 6: back-substitute.

𝑧 = - \frac{1}{11}, 𝑦 = 2 - 7 𝑧 = \frac{29}{11}, 𝑥 = 4 - 0.5 𝑦 - 0.5 𝑧 = \frac{31}{11}

REF is not unique. The stronger RREF (Reduced Row Echelon Form) — pivots equal $1$ and zeros above as well as below — is unique.