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.

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.

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}