Gauss-Jordan Elimination Calculator

Solve systems of linear equations by transforming an augmented matrix into reduced row echelon form.

Enter the coefficients of your linear system and the constant terms to find the unique solution, or determine if no solution or infinite solutions exist.

Examples

Click on any example to load it into the calculator.

2x2 System with a Unique Solution

matrix

A simple 2-variable system that resolves to a single unique solution.

System Size: 2x2

219
1-13

3x3 System with a Unique Solution

matrix

A standard 3-variable system to demonstrate the process on a slightly larger scale.

System Size: 3x3

1129
24-31
36-50

System with No Solution

matrix

An inconsistent system where the row reduction leads to a contradiction (e.g., 0 = 1).

System Size: 3x3

12-14
2529
1336

System with Infinite Solutions

matrix

A dependent system where one equation is a combination of others, leading to a free variable.

System Size: 3x3

1236
25815
1359
Other Titles
Understanding Gauss-Jordan Elimination: A Comprehensive Guide
Master the art of solving linear systems, finding matrix inverses, and understanding vector spaces through the powerful technique of Gauss-Jordan elimination.

What is Gauss-Jordan Elimination?

  • A systematic algorithm for solving systems of linear equations.
  • Transforms a system's augmented matrix into reduced row echelon form.
  • Reveals whether the system has a unique solution, no solution, or infinitely many solutions.
Gauss-Jordan elimination is a cornerstone of linear algebra used to solve systems of linear equations. The method is named after mathematicians Carl Friedrich Gauss and Wilhelm Jordan. It works by systematically performing a sequence of elementary row operations on the augmented matrix of a linear system until it is in a special, simplified form known as reduced row echelon form (RREF).
The Augmented Matrix
A system of linear equations like:
2x + y = 9
x - y = 3
can be represented by an augmented matrix. This is a matrix containing the coefficients of the variables and the constant terms, separated by a vertical line:
[ 2 1 | 9 ]
[ 1 -1 | 3 ]
The Goal: Reduced Row Echelon Form (RREF)
The objective is to transform this augmented matrix into RREF, which has three main properties:
1. The first non-zero element in each non-zero row (the leading entry or pivot) is 1.
2. Each leading 1 is the only non-zero entry in its column.
3. All zero rows are at the bottom of the matrix.
For our example, the RREF would be:
[ 1 0 | 4 ]
[ 0 1 | 1 ]
This directly gives the solution: x = 4, y = 1.

Step-by-Step Guide to Using the Calculator

  • Select the size of your linear system.
  • Input the coefficients into the augmented matrix.
  • Interpret the results correctly for all solution types.
Our Gauss-Jordan Elimination Calculator simplifies this process into a few easy steps.
1. Define Matrix Size
Start by selecting the number of equations (rows) and variables (columns) in your system. The calculator supports various sizes to handle both simple and complex problems.
2. Populate the Augmented Matrix
Enter the coefficients of each variable into the main part of the matrix (A). Then, enter the constant terms from the right-hand side of each equation into the last column (b). Ensure each number is entered correctly.
3. Calculate and Analyze
Click the 'Calculate' button. The tool will perform the elementary row operations and display the final matrix in reduced row echelon form, along with the solution.
- Unique Solution: The RREF will be an identity matrix on the left, with the solution in the final column.
- No Solution: You will see a row with all zeros on the left and a non-zero entry on the right (e.g., [0 0 0 | 1]), which represents a contradiction like 0 = 1.
- Infinite Solutions: You will have fewer non-zero rows than variables, indicating the presence of free variables.

Real-World Applications of Gauss-Jordan Elimination

  • Engineering: Analyzing electrical circuits and structural loads.
  • Computer Science: Solving problems in computer graphics and network flow.
  • Economics: Modeling market equilibrium and optimizing resource allocation.
Gauss-Jordan elimination is not just an academic exercise; it is a powerful tool used to solve tangible problems across many disciplines.
Circuit Analysis (Kirchhoff's Laws)
In electronics, Kirchhoff's laws for current and voltage produce systems of linear equations. Engineers use Gauss-Jordan elimination to solve for the unknown currents in different parts of a complex circuit, which is crucial for designing and troubleshooting electronic devices.
Chemistry
When balancing chemical equations, a system of linear equations can be set up to ensure the number of atoms of each element is conserved. Solving this system gives the stoichiometric coefficients for the reactants and products.
Economics and Finance
Economists use linear systems to model supply and demand, calculate portfolio risk, and analyze input-output models that describe the interdependencies between different sectors of an economy. Gauss-Jordan provides the means to find equilibrium points and optimal strategies.

The Three Elementary Row Operations

  • Swapping two rows.
  • Multiplying a row by a non-zero scalar.
  • Adding a multiple of one row to another row.
The entire Gauss-Jordan elimination algorithm is built upon three simple, yet powerful, operations. These operations allow us to manipulate the matrix without changing the solution set of the underlying linear system.
1. Row Swapping (Ri <-> Rj)
Any two rows in the matrix can be swapped. This is equivalent to changing the order in which you write down the equations, which has no effect on the final solution.
2. Row Scaling (k * Ri -> Ri)
You can multiply any row by a non-zero constant. This corresponds to multiplying both sides of an equation by the same number, which preserves the equality.
3. Row Addition (Ri + k * Rj -> R_i)
A multiple of one row can be added to another row. This is the workhorse of the elimination process. It's like adding one equation (or a multiple of it) to another, a standard technique for solving systems of equations.

Gaussian Elimination vs. Gauss-Jordan Elimination

  • Gaussian elimination produces a row echelon form.
  • Gauss-Jordan elimination continues to reduced row echelon form.
  • Gauss-Jordan requires more steps but gives the solution directly.
While closely related, there is a key distinction between Gaussian elimination and Gauss-Jordan elimination.
Gaussian Elimination: The First Phase
Gaussian elimination transforms the augmented matrix into row echelon form (REF). In REF, the leading entries are 1, and any entries below the leading entry in the same column are zero. However, entries above the leading entry are not necessarily zero. Once in REF, the solution is found using a process called back substitution.
Gauss-Jordan Elimination: The Full Reduction
Gauss-Jordan elimination takes it a step further. After reaching row echelon form, it proceeds with a 'backwards' phase to eliminate the non-zero entries above the leading 1s. This results in the reduced row echelon form (RREF), where each leading 1 is the only non-zero entry in its column. The primary advantage is that when RREF is achieved, the solution to the system can be read directly from the matrix without needing back substitution.
In summary, Gauss-Jordan is a more complete version of Gaussian elimination. While it may involve more computational steps, its directness makes it a preferred method for both manual and computational solutions.