Linear Algebra · step 2 of 13

Gaussian elimination

Gaussian elimination row reduces the augmented matrix to echelon form, from which the solution set is read directly. Three outcomes are possible: a unique solution (a pivot in every variable column), infinitely many (a free variable), or none (a pivot in the final column, giving a row like 0 = 1). Recognising which case you are in is the main examinable skill.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Build the augmented matrixCoefficients on the left, constants on the right, with a consistent variable order across all rows.
Reduce to row echelon formUse only the three elementary operations: swap rows, scale a row by a non-zero constant, add a multiple of one row to another.
Read the classificationA row 0 0 0 | c with c ≠ 0 means no solution. Otherwise, count pivots: fewer pivots than variables gives free variables.
Back-substitute or parametriseFor a unique solution, back-substitute. For free variables, set them as parameters and express the pivots in terms of them.

Worked example

Solve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.

Augmented rows: (1,1,1|6), (2,−1,1|3), (1,2,−1|2).
R2 − 2R1: (0,−3,−1|−9). R3 − R1: (0,1,−2|−4).
R3 + R2/3: (0,0,−2−1/3|−4−3) = (0,0,−7/3|−7), so z = 3.
Back-substitute into −3y − z = −9: −3y = −6, y = 2. Then x = 6 − 2 − 3 = 1.

Answer. x = 1, y = 2, z = 3 — a unique solution, confirmed by substituting into all three equations.

Where marks get dropped

These are the specific errors that cost credit on gaussian elimination questions — QED's rubric penalises each of them separately.

Using an illegal row operation, such as multiplying a row by zero, which destroys information.
Misreading 0 = 0 as no solution. That row is redundant and signals a free variable; 0 = c with c ≠ 0 is the inconsistent case.
Forgetting to carry the augmented column through every operation.

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Gaussian elimination — frequently asked questions

What is the difference between row echelon and reduced row echelon form?

Echelon form has zeros below each pivot; reduced form also has zeros above and pivots equal to 1. Reduced form is unique, which is why it is used for computing inverses and null spaces.

How many solutions can a linear system have?

Exactly zero, one, or infinitely many — never two or three. That trichotomy follows from the structure of the solution set as a translated subspace.

How do I write an infinite solution set?

Parametrise: set each free variable to a parameter and express the others in terms of them, giving a particular solution plus a span of vectors.

The rest of Linear Algebra

Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.

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