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Gauss-Jordan Elimination

Reduces [A|b] to RREF (reduced row echelon form): pivots = 1, zeros above and below. Reads off solution directly. Same as Gaussian elimination plus backward elimination.

Concept Fundamentals
pivots=1, zeros above/below
RREF
[A|b]
Augmented
from RREF
Solution
swap, scale, add
Row ops

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RREF is unique for each matrix. Solution from last column of RREF. Same as Gaussian + backward pass.

Key quantities
pivots=1, zeros above/below
RREF
Key relation
[A|b]
Augmented
Key relation
from RREF
Solution
Key relation
swap, scale, add
Row ops
Key relation

Ready to run the numbers?

Why: RREF gives solution immediately. Pivot columns = leading variables; free columns = parameters. Foundation for linear algebra.

How: Forward: create zeros below pivots (Gaussian). Backward: create zeros above pivots. Scale rows so pivots = 1.

RREF is unique for each matrix.Solution from last column of RREF.
Sources:Gilbert Strang, Linear AlgebraKhan Academy

Run the calculator when you are ready.

Gauss-Jordan to RREFFull reduction to RREF

Gauss-Jordan Elimination Calculator

Augmented Matrix [A|b]

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Enter the augmented matrix [A|b] for your system. The last column represents the constants.

What is Gauss-Jordan Elimination?

Gauss-Jordan elimination is an algorithm for solving systems of linear equations by transforming the augmented matrix of the system to its Reduced Row Echelon Form (RREF). It's an extension of Gaussian elimination that continues the process until every column with a pivot has zeros in all positions except for the pivot.

The method systematically applies elementary row operations to transform the augmented matrix into RREF:

  1. Swap rows to put the matrix in a form where pivots appear along the diagonal
  2. Scale each pivot row so that the pivot equals 1
  3. Eliminate entries above and below each pivot to get zeros

Once the matrix is in RREF, the solution to the system can be directly read from the matrix, or it may reveal that the system has no solution or infinitely many solutions.

Elementary Row Operations

The Gauss-Jordan elimination method uses three types of elementary row operations, each of which preserves the solution set of the system:

1. Row Swap

Exchange two rows:

Riโ†”RjR_i \leftrightarrow R_j

This operation is useful for positioning nonzero entries in pivot positions.

2. Row Scaling

Multiply a row by a nonzero scalar:

Riโ†’cโ‹…Ri(cโ‰ 0)R_i \rightarrow c \cdot R_i \quad (c \neq 0)

This operation is used to obtain a leading 1 in a pivot position.

3. Row Addition

Add a multiple of one row to another:

Riโ†’Ri+cโ‹…RjR_i \rightarrow R_i + c \cdot R_j

This operation is used to create zeros above and below pivot positions.

Reduced Row Echelon Form (RREF)

A matrix is in Reduced Row Echelon Form (RREF) if it satisfies the following conditions:

1. All rows consisting entirely of zeros are at the bottom of the matrix.

2. The leading entry (first nonzero element) of each nonzero row is 1 (called a pivot).

3. Each leading 1 appears in a column to the right of the leading 1 in the row above it.

4. All entries in a column above and below a leading 1 are zeros.

Example of a matrix in RREF:

(100a010b001c0000)\begin{pmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \\ 0 & 0 & 0 & 0 \end{pmatrix}

Interpreting the Results

Unique Solution

If the RREF has a pivot in every column except the last one (the constant column), and every row has a pivot, then the system has a unique solution. The solution can be read directly from the augmented column.

(100โˆฃ2010โˆฃ3001โˆฃโˆ’1)โ‡’{x1=2x2=3x3=โˆ’1\begin{pmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 3 \\ 0 & 0 & 1 & | & -1 \end{pmatrix} \quad \Rightarrow \quad \begin{cases} x_1 = 2 \\ x_2 = 3 \\ x_3 = -1 \end{cases}

No Solution

If there is a row with all zeros except the last entry (which is nonzero), the system is inconsistent and has no solution. This corresponds to a contradiction like 0 = k (where k โ‰  0).

(10โˆฃ201โˆฃ300โˆฃ5)โ‡’No solution (inconsistent system)\begin{pmatrix} 1 & 0 & | & 2 \\ 0 & 1 & | & 3 \\ 0 & 0 & | & 5 \end{pmatrix} \quad \Rightarrow \quad \text{No solution (inconsistent system)}

Infinitely Many Solutions

If there are fewer pivots than variables, the system has infinitely many solutions. Some variables (the ones without pivots in their columns) are free variables that can take any value. The other variables (basic variables) are expressed in terms of these free variables.

(102โˆฃ4013โˆฃ5000โˆฃ0)โ‡’{x1=4โˆ’2x3x2=5โˆ’3x3x3 is a free variable\begin{pmatrix} 1 & 0 & 2 & | & 4 \\ 0 & 1 & 3 & | & 5 \\ 0 & 0 & 0 & | & 0 \end{pmatrix} \quad \Rightarrow \quad \begin{cases} x_1 = 4 - 2x_3 \\ x_2 = 5 - 3x_3 \\ x_3 \text{ is a free variable} \end{cases}

Applications of Gauss-Jordan Elimination

Solving Linear Systems

The primary application of Gauss-Jordan elimination is solving systems of linear equations in various fields:

  • Engineering: Circuit analysis, structural analysis, and control systems
  • Economics: Input-output models, equilibrium analysis
  • Computer graphics: Transformations, camera calibration
  • Physics: Finding equilibrium states, analyzing constraints

Matrix Inversion

Gauss-Jordan elimination can be used to find the inverse of a matrix. By augmenting the matrix with the identity matrix and performing row operations to transform the original matrix to the identity matrix, the augmented part becomes the inverse.

(AโˆฃI)โ†’row ops(IโˆฃAโˆ’1)\begin{pmatrix} A & | & I \end{pmatrix} \xrightarrow{\text{row ops}} \begin{pmatrix} I & | & A^{-1} \end{pmatrix}

Calculating Matrix Rank

The rank of a matrix is equal to the number of nonzero rows in its row echelon form, which can be computed using Gauss-Jordan elimination. This is useful in determining the dimension of the column space and nullspace of a matrix.

Computational Efficiency

While Gauss-Jordan elimination is conceptually clear, for large systems, other methods like LU decomposition may be more computationally efficient. However, for small to medium-sized systems, Gauss-Jordan elimination provides a direct and comprehensive solution technique.

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

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RREF unique

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Pivots = 1

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