Understand the subject
What is Gaussian elimination?
Gaussian elimination uses reversible row operations to expose pivots below a staircase of zeros.
Use it to solve systems by back substitution and inspect rank structure.
See the structure
What the calculation is doing
[ 1 1 | 3 ]R₂ − 2R₁[ 2 1 | 4 ]→pivots reveal the solution
Worked interpretation
Read the result in context
[A|b] → row echelon form
Interpret with care
Important boundary
Different valid row-operation paths can lead to different echelon forms but the same solution set.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Set the matrix dimensions required by the task.
- Fill every visible entry and any operation-specific control.
- Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
Use row swaps and row replacements to clear entries below each pivot.
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
The resulting echelon form exposes pivots for back substitution.
Use row swaps and row replacements to clear entries below each pivot.
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Exactness
Finite decimals become reduced rational numbers and intermediate structural calculations are not rounded.
