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Gauss-Jordan Elimination Calculator

Solve linear systems and find reduced row echelon form with step-by-step solutions

Matrix Input

Choose matrix size. Use augmented matrices (like 3×4) for solving linear systems.

Enter real numbers for matrix elements

Results

3
Matrix Rank
Yes
In RREF
unique
Solution Type

Reduced Row Echelon Form

|1.0000013.000|
|01.0000-8.000|
|001.000-2.000|

Step-by-Step Solution

Step 0: Initial Matrix

Operation

Starting with the given matrix

|1.002.00-2.001.00|
|2.004.00-5.004.00|
|2.002.003.004.00|

Step 1: R1 ↔ R2

Operation

Swap row 1 with row 2 to get pivot

|2.004.00-5.004.00|
|1.002.00-2.001.00|
|2.002.003.004.00|

Step 2: R1 → (1/2.000) × R1

Operation

Make pivot in row 1, column 1 equal to 1

|1.002.00-2.502.00|
|1.002.00-2.001.00|
|2.002.003.004.00|

Step 3: R2 → R2 + (-1.000) × R1

Operation

Eliminate entry in row 2, column 1

|1.002.00-2.502.00|
|000.50-1.00|
|2.002.003.004.00|

Step 4: R3 → R3 + (-2.000) × R1

Operation

Eliminate entry in row 3, column 1

|1.002.00-2.502.00|
|000.50-1.00|
|0-2.008.000|

Step 5: R2 ↔ R3

Operation

Swap row 2 with row 3 to get pivot

|1.002.00-2.502.00|
|0-2.008.000|
|000.50-1.00|

Step 6: R2 → (1/-2.000) × R2

Operation

Make pivot in row 2, column 2 equal to 1

|1.002.00-2.502.00|
|01.00-4.000|
|000.50-1.00|

Step 7: R1 → R1 + (-2.000) × R2

Operation

Eliminate entry in row 1, column 2

|1.0005.502.00|
|01.00-4.000|
|000.50-1.00|

Step 8: R3 → (1/0.500) × R3

Operation

Make pivot in row 3, column 3 equal to 1

|1.0005.502.00|
|01.00-4.000|
|001.00-2.00|

Step 9: R1 → R1 + (-5.500) × R3

Operation

Eliminate entry in row 1, column 3

|1.000013.00|
|01.00-4.000|
|001.00-2.00|

Step 10: R2 → R2 + (4.000) × R3

Operation

Eliminate entry in row 2, column 3

|1.000013.00|
|01.000-8.00|
|001.00-2.00|

Elementary Row Operations

1. Row Swapping: R₁ ↔ R₂ (interchange two rows)

2. Row Scaling: R₁ → c × R₁ (multiply a row by non-zero constant)

3. Row Addition: R₁ → R₁ + c × R₂ (add multiple of one row to another)

Example: Solving Linear System

System of Equations

x + 2y - 2z = 1

2x + 4y - 5z = 4

2x + 2y + 3z = 4

Augmented Matrix

| 1 2 -2 | 1 |

| 2 4 -5 | 4 |

| 2 2 3 | 4 |

Solution

x = 13, y = -8, z = -2

Unique solution found through Gauss-Jordan elimination

Key Concepts

1

RREF

Reduced Row Echelon Form

→

Row Operations

Elementary transformations

R

Rank

Number of non-zero rows

Applications

•

Solving linear systems

•

Finding matrix inverse

•

Computing matrix rank

•

Linear independence testing

•

Basis computation

Understanding Gauss-Jordan Elimination

What is Gauss-Jordan Elimination?

Gauss-Jordan elimination is an algorithm that uses elementary row operations to transform a matrix into reduced row echelon form (RREF). This method is used to solve systems of linear equations, find matrix inverses, and determine matrix properties.

Row Echelon Form vs RREF

  • • REF: Leading entries in descending staircase pattern
  • • RREF: Leading entries are 1, with zeros above and below
  • • Uniqueness: RREF is unique, REF is not

Algorithm Steps

  1. Find pivot (leftmost non-zero entry)
  2. Swap rows to move pivot to top
  3. Scale row to make pivot equal 1
  4. Use row operations to make other entries in column zero
  5. Repeat for next column and row

Note: The process continues until all possible leading 1's are created and all entries above and below them are eliminated.

Related Linear Algebra Calculators

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Matrix Rank Calculator

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