Gaussian Elimination

Reduced Row Echelon Form

Reduce a 3x3 augmented matrix to reduced row echelon form (RREF). Forward elimination creates the upper-triangular REF; back- substitution then normalizes each pivot to 1 and zeros entries above each pivot, yielding a unique solution directly.

rref A matrix is in reduced row echelon form when it is in REF and every pivot equals 1 with zeros both above and below it.
back-substitution Starting from the bottom-right pivot, divide each pivot row to make the pivot 1, then subtract multiples from rows above to eliminate that column entry everywhere else.

Example

Continue row operations until each pivot column is reduced.

highlighted = computed this step

Step 1 — Set up

Start with the augmented matrix.

[111323481024]\left[\begin{array}{ccc|c}1&1&1&3\\2&3&4&8\\1&0&2&4\end{array}\right]

Step 2 — Update row 2

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

[111301221024]\left[\begin{array}{ccc|c}1&1&1&3\\\hl{0}&\hl{1}&\hl{2}&\hl{2}\\1&0&2&4\end{array}\right]

Step 3 — Update row 3

Update row 3 using row 1: r3 <- r3 - r1.

[111301220-111]\left[\begin{array}{ccc|c}1&1&1&3\\0&1&2&2\\\hl{0}&\hl{-1}&\hl{1}&\hl{1}\end{array}\right]

Step 4 — Update row 3

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

[111301220033]\left[\begin{array}{ccc|c}1&1&1&3\\0&1&2&2\\\hl{0}&\hl{0}&\hl{3}&\hl{3}\end{array}\right]

Step 5 — Normalize row 3

Scale row 3: r3 <- 1/3*r3.

[111301220011]\left[\begin{array}{ccc|c}1&1&1&3\\0&1&2&2\\\hl{0}&\hl{0}&\hl{1}&\hl{1}\end{array}\right]

Step 6 — Back-sub row 1

Update row 1 using row 3: r1 <- r1 - r3.

[110201220011]\left[\begin{array}{ccc|c}\hl{1}&\hl{1}&\hl{0}&\hl{2}\\0&1&2&2\\0&0&1&1\end{array}\right]

Step 7 — Back-sub row 2

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

[110201000011]\left[\begin{array}{ccc|c}1&1&0&2\\\hl{0}&\hl{1}&\hl{0}&\hl{0}\\0&0&1&1\end{array}\right]

Step 8 — Back-sub row 1

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

[100201000011]\left[\begin{array}{ccc|c}\hl{1}&\hl{0}&\hl{0}&\hl{2}\\0&1&0&0\\0&0&1&1\end{array}\right]

Step 9 — Result

Read the solution: x1 = 2, x2 = 0, x3 = 1.

[100201000011],solution: x1=2, x2=0, x3=1\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&0\\0&0&1&1\end{array}\right],\quad \text{solution: }x1=2,\ x2=0,\ x3=1