Gaussian Elimination

Row Echelon Form

Reduce a 3x3 augmented matrix to row echelon form using forward elimination only. Each row operation zeros out entries below the pivot, creating an upper-triangular staircase pattern. Back- substitution is not performed — the goal is REF recognition.

Example

Eliminate entries below each pivot to reach row echelon form.

highlighted = computed this step

Step 1 — Set up

Start with the augmented matrix.

[1236257130124]\left[\begin{array}{ccc|c}1&2&3&6\\2&5&7&13\\0&1&2&4\end{array}\right]

Step 2 — Update row 2

Update row 2 using row 1: r2 <- r2 - 2*r1.

[123601110124]\left[\begin{array}{ccc|c}1&2&3&6\\\hl{0}&\hl{1}&\hl{1}&\hl{1}\\0&1&2&4\end{array}\right]

Step 3 — Update row 3

Update row 3 using row 2: r3 <- r3 - r2.

[123601110013]\left[\begin{array}{ccc|c}1&2&3&6\\0&1&1&1\\\hl{0}&\hl{0}&\hl{1}&\hl{3}\end{array}\right]

Step 4 — Result

Read the row echelon form.

[123601110013],row echelon form reached\left[\begin{array}{ccc|c}1&2&3&6\\0&1&1&1\\0&0&1&3\end{array}\right],\quad \text{row echelon form reached}
row-echelon-form A matrix is in row echelon form when each pivot is strictly to the right of the pivot in the row above, and all entries below each pivot are zero.
forward-elimination Use row operations to zero all entries below each pivot column by column from left to right. Stop when the matrix is upper triangular.