JENECONKMaths Universe

Row Reduction and Gaussian Elimination

Ages 14–15

Difficulty adapts to you

Row Reduction and Gaussian Elimination

Row operations preserve the solutions of a system when applied to an augmented matrix. You may swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. Use these operations to create zeros below pivots, then back-substitute. For a system, the augmented matrix keeps coefficients and constants together; for an inverse, reduce [A | I] to [I | A⁻¹].

Your goals

  • ✓You can use elementary row operations to simplify matrices and solve systems.
  • ✓You can use pivot, augmented matrix, row operation correctly.
  • ✓You can apply every row operation to the entire row and read contradiction rows carefully.

Words to know

pivot:
A leading non-zero entry used to eliminate values in a column.
augmented matrix:
A coefficient matrix joined to its constants column.
row operation:
A legal swap, scale, or row replacement that preserves solutions.

Examples

  • 1The equations x + y = 5 and 2x + y = 8 become 115218. Replace row 2 by row 2 − 2(row 1) to get 1150−1−2.
  • 2A row swap is written R₁ ↔ R₂; scaling is R₂ → 3R₂; replacing a row is R₂ → R₂ − 2R₁.
  • 3A row [0, 0, 4] in an augmented matrix means 0 = 4, so the system is inconsistent and has no solution.
  • 4A row [0, 0, 0] gives no new restriction; it may indicate infinitely many solutions when another free variable remains.
  • 5Keep the same operation on every entry in a row. Changing only one coefficient changes the system and invalidates the result.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Forward elimination

Best for: Solving a system systematically
  1. 1.Choose a non-zero pivot.
  2. 2.Use it to create zeros below it.
  3. 3.Move to the next column and repeat.

Example: R₂ → R₂ − 2R₁ removes the first variable from row 2.

Method 2

Augmented-matrix reading

Best for: Recognising solution types
  1. 1.Separate coefficient columns from the constants column.
  2. 2.Read any zero rows or contradictions.
  3. 3.Back-substitute only after the pivot structure is clear.

Example: [0, 0 | 4] means 0 = 4, so the system has no solution.

Check these

  • • Do not change one entry instead of the whole row or divide by a zero pivot.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own row reduction and gaussian elimination example, solve it, and explain how you checked it.

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Helpful Hints

“Look for patterns — maths is full of them!”