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Gaussian Elimination

Gaussian elimination solver. Enter A and b to walk through row operations step by step and back-substitute—for linear systems.

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[A|b] → echelon form → back substitute
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4
space-separated

📖 Tutorial｜Gaussian Elimination

1. Definition
Gaussian elimination uses elementary row operations to reduce the augmented matrix to echelon form, then back-substitutes.
2. Symbols
SymbolMeaning
pivotlargest entry in a column
echelon formleading entries move right each row
back substitutionsolve bottom-up
3. How this tool works
  • Partial pivoting improves numerical stability;
  • Row ops: scale a row, swap rows, add a multiple of one row to another;
  • After echelon form, solve bottom-up.
4. Steps
  1. Enter A and b;
  2. Click Calculate to see each row operation;
  3. Read the back-substituted solution.
5. Example
Example: A=1 1 1;2 1 −1;−1 1 2, b=6 1 5. Elimination gives x₁=1, x₂=2, x₃=3.
6. Pitfalls
Swap rows when the pivot is near zero;
Row operations preserve the solution;
No solution shows a 0=nonzero row.

❓ FAQ｜Gaussian Elimination

Three steps of elimination?
Pivot, eliminate, back substitute.
What is a pivot?
The largest entry in the current column, used to stabilize elimination.
What does echelon form look like?
Leading entries move right each row, zeros below.
Why pivot?
To avoid pivots near zero causing numerical instability.
Elimination vs Cramer?
Elimination O(n³) beats Cramer O(n!) for large systems.