Gaussian Elimination
Reduce a matrix to row echelon form (REF) using Gaussian elimination. Get instant step-by-step solutions — completely free, no signup required.
Free Gaussian Elimination Calculator
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About Gaussian Elimination
Reduce a matrix to row echelon form (REF) using Gaussian elimination. Linear algebra and matrix operations are foundational to computer graphics, machine learning, and solving systems of equations. Understanding matrix manipulations is critical for advanced mathematics.
Our Gaussian Elimination calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any gaussian elimination expression and receive a complete, verified solution with every step shown.
Gaussian Elimination Formula
Row operations: Rᵢ ↔ Rⱼ, kRᵢ, Rᵢ + kRⱼ
Gaussian elimination solves a system of linear equations by using row operations on the augmented matrix to reach row echelon form — zeros below each leading entry. Back-substitution from the bottom row then gives each variable.
Worked Example
Solve x + y + z = 6, 2x + 3y + z = 14, x − y + 2z = 5
- R2 − 2R1 → [0, 1, −1 | 2]; R3 − R1 → [0, −2, 1 | −1]
- R3 + 2R2 → [0, 0, −1 | 3]
- Back-substitute: z = −3
- y − z = 2 → y = −1; x + y + z = 6 → x = 10
Answer: x = 10, y = −1, z = −3
Common Mistakes to Avoid
- Applying an operation to the coefficients but not the augmented column.
- Dividing by a zero pivot instead of swapping rows first.
- Making arithmetic slips with negatives — check by substituting into the original equations.
How to Use the Gaussian Elimination Calculator
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Input your matrix or vector values for the Gaussian Elimination problem
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Press Enter or click the Solve button to run the calculation
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Review the step-by-step solution that appears below the input
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Check the breakdown of each operation to understand the method
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Try a different problem to practice and build your understanding
Frequently Asked Questions
What is the Gaussian Elimination formula?
Row operations: Rᵢ ↔ Rⱼ, kRᵢ, Rᵢ + kRⱼ. Gaussian elimination solves a system of linear equations by using row operations on the augmented matrix to reach row echelon form — zeros below each leading entry. Back-substitution from the bottom row then gives each variable.
Can you show a worked Gaussian Elimination example?
Solve x + y + z = 6, 2x + 3y + z = 14, x − y + 2z = 5: R2 − 2R1 → [0, 1, −1 | 2]; R3 − R1 → [0, −2, 1 | −1]; R3 + 2R2 → [0, 0, −1 | 3]; Back-substitute: z = −3; y − z = 2 → y = −1; x + y + z = 6 → x = 10. Answer: x = 10, y = −1, z = −3.
Is this Gaussian Elimination calculator free to use?
Yes — the Gaussian Elimination calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.
Can I use this for Gaussian Elimination homework and exam practice?
Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.