Gauss-Jordan Calculator

Perform Gauss-Jordan elimination on augmented matrices [A|b]. Solve linear systems, find matrix inverses by augmenting with I, compare with Gaussian elimination, and understand partial pivoting and O(n³) complexity.

Solution (x1, x2, x3)
—
Final RREF —
Rank —
Extended More scenarios, charts & detailed breakdown ▾
Solution
—
Augmented RREF —
Professional Full parameters & maximum detail ▾

Solution & RREF

Solution —
RREF —

Algorithm Comparison

G-J vs Gaussian elimination —
Partial pivoting note —
Complexity —

How to Use This Calculator

  1. Enter augmented matrix [A|b] entries — solution and RREF appear instantly.
  2. Use Find Inverse tab to augment with I and extract A-inverse.
  3. Use Professional to compare with Gaussian elimination and learn about pivoting.

Formula

Row operations: (1) Swap rows, (2) Scale row by nonzero scalar, (3) Add multiple of one row to another.

Goal: produce RREF where pivot columns = identity.

Example

[[2,1,-1|8],[-3,-1,2|-11],[-2,1,2|-3]] → RREF → x1=2, x2=3, x3=-1.

Frequently Asked Questions

  • An algorithm that applies row operations to an augmented matrix [A|b] to produce RREF. Unlike Gaussian elimination (which stops at REF), Gauss-Jordan fully reduces so the solution can be read directly.
  • Augment A with the identity [A|I]. Apply row operations until the left side becomes I. The right side is then A-inverse. This uses 2n³/3 operations for an n×n matrix.
  • At each column, swap rows so the largest absolute value is the pivot. This prevents division by small numbers and reduces numerical error.
  • Gaussian elimination produces Row Echelon Form (zeros below pivots) then uses back-substitution. Gauss-Jordan goes further to RREF (zeros above and below each pivot), reading solutions directly.
  • O(n³) for an n×n system. LU decomposition is preferred when solving Ax=b for many different b vectors, since it factorizes A once and solves each b in O(n²).

Related Calculators

rref RREF Calculator A-1 Matrix Inverse |A| Determinant Calculator [] Matrix Calculator
Sources & References (5) ▾
  1. Introduction to Linear Algebra — Gilbert Strang — Wellesley-Cambridge Press
  2. OpenStax College Algebra — Gaussian Elimination — OpenStax
  3. MIT OCW 18.06 — Elimination with Matrices — MIT OpenCourseWare
  4. Khan Academy — Reduced Row Echelon Form — Khan Academy
  5. Numerical Recipes — LU Decomposition and Linear Systems — Cambridge University Press