Gauss-Jordan Elimination Calculator

The Gauss-Jordan Elimination Algorithm

Named after mathematicians Carl Friedrich Gauss and Wilhelm Jordan, the Gauss-Jordan elimination algorithm is a systematic method for solving systems of linear equations and manipulating matrices. While Gaussian elimination stops at an upper triangular matrix (Row Echelon Form), Gauss-Jordan goes a step further to yield the unique Reduced Row Echelon Form (RREF).

The algorithm consists of two main phases:

Forward Phase (Gaussian Elimination): We work from left to right, column by column. In each column, we select a row with a non-zero entry in that column to act as our pivot. We swap this row to the top, scale the pivot to 1, and subtract multiples of this row from all rows *below* it to clear out the column. This results in Row Echelon Form (REF).
Backward Phase (Jordan Elimination): We work from right to left, bottom to top. For each pivot, we subtract multiples of its row from the rows *above* it to turn all entries in the pivot's column to zero. This results in Reduced Row Echelon Form (RREF).

The Three Elementary Row Operations

Gauss-Jordan elimination is executed entirely using three types of operations that preserve the relationships between matrix rows:

Row Swap: Exchanging the positions of two rows (written as R_i ↔ R_j). This is used when a zero is present at the pivot position.
Row Scaling: Multiplying a row by a non-zero scalar (written as R_i → cR_i). This is used to scale a leading entry to 1.
Row Addition/Subtraction: Adding or subtracting a multiple of one row to another row (written as R_i → R_i + cR_j). This is used to eliminate non-zero entries in a pivot's column.

Why Use Gauss-Jordan Elimination?

Although numerical computing packages often use LU decomposition for efficiency, Gauss-Jordan remains the standard teaching tool because it represents the most direct way to resolve a matrix into its constituent spaces. It provides a clear, algebraic representation of the relationships between vectors and shows whether a system of equations has a unique solution, no solution, or infinitely many solutions.

Frequently Asked Questions

How does Gauss-Jordan elimination work?

Gauss-Jordan elimination is an extension of Gaussian elimination. While Gaussian elimination reduces a matrix to upper triangular (REF), Gauss-Jordan continues to eliminate entries above the pivots as well, yielding the unique RREF.

What are elementary row operations?

Elementary row operations are three types of operations that can be performed on a matrix without changing the solution set of the system it represents: 1. Swapping two rows. 2. Scaling a row by a non-zero constant. 3. Adding or subtracting a multiple of one row to another.

How do you calculate RREF by hand?

To calculate RREF by hand, use Gauss-Jordan elimination: 1. Find the first column with a non-zero entry. 2. Swap rows if needed to put a non-zero entry at the top (the pivot). 3. Scale the pivot row to make the pivot 1. 4. Add multiples of the pivot row to all other rows to eliminate the entries in that column. 5. Repeat for the remaining rows and columns.

What is a pivot in RREF?

A pivot (or leading entry) is the first non-zero number in a row of a matrix. When a matrix is in RREF, every pivot must be equal to 1, and it must be the only non-zero entry in its column.

Matrix A

Matrix B

Step-by-Step Row Working