Gaussian Elimination

Systematic row reduction to solve linear systems, compute rank, and find inverses — the foundational algorithm of linear algebra.

Gaussian elimination — 2x+y-z=8, -3x-y+2z=-11, -2x+y+2z=-3

pivot row target row

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Initial augmented matrix [A|b]

Step 0 of 4

Definition

Gaussian elimination is the standard algorithm for solving systems of linear equations $A\mathbf{x} = \mathbf{b}$ .

It transforms the augmented matrix $[A|\mathbf{b}]$ into row echelon form using three elementary row operations:

Then back substitution solves the triangular system from bottom to top.

Key properties

Each elementary row operation preserves the solution set of the system
Row echelon form has all leading entries (pivots) strictly to the right as you move down rows
The number of pivots equals the rank of the matrix
A row of all zeros equal to a nonzero constant signals an inconsistent (unsolvable) system

Common mistakes

Forgetting to apply a row operation to the entire row (including the augmented column $\mathbf{b}$ ) — a common source of silent arithmetic errors
Dividing by a zero or near-zero pivot: this either breaks the algorithm outright or (if the pivot is merely tiny) introduces large numerical error — partial pivoting exists specifically to avoid this

Solving a 3×3 system

$\begin{pmatrix}2&1&-1\\-3&-1&2\\-2&1&2\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}8\\-11\\-3\end{pmatrix}$

After elimination: upper triangular system → $z=1, y=-2, x=2$ .

Try it

Solve the system: $x + 2y = 5$ , $3x + 4y = 11$ using Gaussian elimination.

Solution

Augmented matrix: $\begin{pmatrix}1&2&5\\3&4&11\end{pmatrix}$

$R_2 \leftarrow R_2 - 3R_1$ : $\begin{pmatrix}1&2&5\\0&-2&-4\end{pmatrix}$

Back substitute: $-2y = -4 \Rightarrow y=2$ . Then $x + 4 = 5 \Rightarrow x=1$ .

Related concepts

Needs first