Gauss Jordan Elimination Calculator - Linear Systems

A Gauss Jordan elimination calculator is a linear algebra tool that takes the coefficients and constants of a linear system, runs Gauss-Jordan elimination to drive the augmented matrix to reduced row echelon form, and returns the solution vector, system status, and ranks for 2x2 and 3x3 systems in a single pass. It is the same routine taught in undergraduate linear algebra, condensed into a small input grid and a results panel that shows the status, the solved unknowns, the rank of the coefficient matrix, and the rank of the augmented matrix side by side.

• • Solve small linear systems: Resolve two-equation and three-equation systems by hand without expanding every cofactor or running Gaussian elimination twice.
• • Check homework and exam answers: Drop the coefficients in, confirm the solution, and read off the rank of A and the augmented matrix to back up your steps.
• • Diagnose singular or inconsistent systems: See at a glance whether rank(A) equals rank([A|b]) so you can tell a unique solution, an infinite family, or a contradiction apart.

Row reduction is the workhorse of linear algebra because it answers three questions at once: does the system have a solution, is the solution unique, and what is it. The calculator keeps that workflow visible by showing the rank, the status, and the solution vector side by side.

If you prefer the half-reduction workflow that still needs a back-substitution pass, Elimination Method Calculator handles the same 2x2 and 3x3 systems using Gaussian elimination.

The calculator builds the augmented matrix from your inputs, then runs Gauss-Jordan elimination: for each pivot column it partial-pivots onto the largest absolute entry, scales the pivot row to a leading 1, and eliminates that column in every other row until the left block is the identity matrix.

• A: Square coefficient matrix of size n x n (2x2 or 3x3 in this calculator)
• b: Right-hand side column vector of length n
• [A | b]: Augmented matrix that appends b as an extra column on the right of A
• I: Identity matrix of the same size as A
• x: Solution vector with one entry per unknown

The augmented matrix is the bridge between the system and the row operations. The three allowed moves are row swaps, scaling a row by a non-zero scalar, and adding a scalar multiple of one row to another. None of these change the solution set, which is why the final RREF still encodes the same linear system as the original inputs.

According to Wikipedia, Gauss-Jordan elimination, Gauss-Jordan elimination is a finite sequence of elementary row operations that reduces an augmented matrix to reduced row echelon form, at which point the rightmost column directly gives the solution of the linear system.

When the system is a 2x2 or 3x3 with a unique solution and you just need the answer fast, System of Equations Calculator returns the same vector using substitution or elimination without showing the row operations.

Four linear-algebra ideas appear every time you run a Gauss Jordan elimination calculator, and they explain why the answer is what it is.

These concepts are not optional: the rank decides the system status, the RREF decides the solution, and the elementary row operations are the only moves that preserve the solution set while you reshape the matrix.

The adjugate and the cofactor matrix are the alternative way to invert a 2x2 or 3x3 matrix, and Adjoint Matrix Calculator returns the cofactor expansion and the inverse matrix in one pass.

The grid is laid out the same way the augmented matrix is written, so typing coefficients into it should feel natural.

1. 1 Pick a matrix size: Use the size selector to choose 2x2 for two equations in two unknowns or 3x3 for three equations in three unknowns.
2. 2 Type the coefficients: Fill the a, b, c (and for 3x3 also d) inputs for each row with the coefficients of the unknowns in the same order they appear in your equations.
3. 3 Add the constants: The last column holds the right-hand side of every equation. For 2x2 that is c1 and c2; for 3x3 that is d1, d2, and d3.
4. 4 Read the status banner: Look at the system status row first. Unique solution, no solution, or infinite solutions tells you how to interpret the values that follow.
5. 5 Read the solution vector: x1, x2, and x3 give the values of the unknowns. When the status is not unique, the values are reported as 0 and the rank of A and rank of [A | b] above explain why.

Type the 2x2 system x1 + x2 = 5 and 2x1 - x2 = 4. The status banner shows Unique solution, x1 reads 3, and x2 reads 2, matching the worked example above.

Once you know the solution to the equality system, Linear Inequality Calculator helps you explore the same linear expressions as inequality constraints in 2D or 3D.

The calculator pays off in five concrete ways when you need to solve a small linear system quickly.

• • Single pass to the answer: Row reduction handles every step from the raw coefficients to the solved vector, so you do not have to repeat Gaussian elimination and back-substitution by hand.
• • Singular cases are flagged: The status banner distinguishes a unique solution, a contradiction, and an infinite family, so you know when to look for a free parameter or a typo in the inputs.
• • Rank readout explains why: rank(A) and rank([A | b]) are reported next to the status, so a singular or inconsistent system is traceable to a specific linear dependency or right-hand side mismatch.
• • Numerical safety from partial pivoting: The routine swaps in the row with the largest absolute pivot, which keeps the elimination stable when a small pivot would otherwise magnify rounding error.
• • Real-time recalculation: Inputs trigger an instant update, so you can sweep a coefficient to see how the solution vector responds without clicking a Calculate button.

For a 2x2 or 3x3 system, the answers are usually small enough to check by hand, which is why the calculator is useful as a verification step before you commit to a result.

When the system comes from a least-squares fit rather than an exact equality, Linear Regression Calculator solves the normal equations A^T A beta = A^T y for the slope and intercept.

Four things decide what comes out of the calculator, and a couple of them limit how far the method can be pushed.

• • This page is sized for 2x2 and 3x3 systems. For 4x4 and larger, the form layout would need a denser grid and Gaussian elimination with back-substitution is usually a more efficient workflow, since the result row would have to show four unknowns at once.
• • Floating-point rounding can leave very small residual entries (around 1e-15) inside the routine instead of an exact zero. The calculator treats anything below 1e-10 as zero for the rank count, so the rank readout stays stable.

Treat the system status and the two rank readouts as the authority on whether a solution exists, and treat the values of x1, x2, and x3 as the authority on what the solution is.

As published by Wolfram MathWorld, Gauss-Jordan elimination, the reduced row echelon form of a matrix is unique, which is why Gauss-Jordan elimination always produces the same final system regardless of the order of the elementary row operations used.