Gaussian Elimination Calculator
Solve a 2x2 or 3x3 linear system by row-reducing the augmented matrix to reduced row-echelon form, with every elementary row operation shown.
Gaussian Elimination Calculator
Calculator
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Formula: Row Reduce [A|b] to RREF using: R_i ↔ R_j, R_i → kR_i, R_i → R_i + kR_j
Worked example — x = 1.75, y = 1.5
Formula
Row Reduce [A|b] to RREF using: R_i ↔ R_j, R_i → kR_i, R_i → R_i + kR_j
Gaussian elimination converts the augmented matrix [A|b] to row-echelon form (REF) or reduced row-echelon form (RREF) using elementary row operations. Back-substitution then gives the solution.
Worked Examples
Example 1: 2x2 system
Problem:2x + y = 5, 4x - 6y = -2
Solution:Augmented: [2,1,5;4,-6,-2]. Row2 - 2×Row1 gives -8y = -12, so y = 1.5 and x = (5 - 1.5)/2 = 1.75
Result:x = 1.75, y = 1.5
Frequently Asked Questions
What if the system has no solution?
If a row becomes [0 0 ... 0 | nonzero], the system is inconsistent (no solution). If a row becomes all zeros, the system has infinitely many solutions (underdetermined).
How do I enter the augmented matrix?
Enter each row separated by semicolons, with commas between values. The last value in each row is the constant. Example: "2,1,5;4,-6,-2" means 2x+y=5 and 4x-6y=-2.
Why does the calculator use partial pivoting?
Partial pivoting swaps rows so that the largest absolute value in each column is used as the pivot. This reduces numerical errors caused by dividing by very small numbers. Without pivoting, floating-point rounding can make results wildly inaccurate even for well-conditioned systems.
What are real-world applications of Gaussian elimination?
Gaussian elimination is used in electrical circuit analysis (finding currents via Kirchhoff's laws), structural engineering (solving force equilibrium equations), computer graphics (transformations and lighting calculations), economics (input-output models), and virtually any engineering or science problem that requires solving simultaneous linear equations.
What is the difference between row echelon form and reduced row echelon form?
Row echelon form only requires a staircase of leading entries with zeros below each pivot, after which you finish by back-substitution. Reduced row echelon form goes further and clears the entries above each pivot as well, leaving an identity block whose last column is the solution. Gaussian Elimination Calculator eliminates in both directions, so it is really performing Gauss-Jordan elimination; it does not rescale the pivots to 1, so it finishes by dividing each row constant by its own pivot.
What do the three elementary row operations do to the solution set?
Swapping two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another all leave the solution set unchanged, because each is reversible and each corresponds to restating one equation in an equivalent way. The nonzero restriction matters: scaling a row by zero would destroy an equation and manufacture solutions that the original system never had. That is why elimination can be done in any order and still land on the same answer.
Why does a zero pivot mean there is no unique solution?
A pivot column that is entirely zero from the current row down means no equation left in the system contributes new information about that variable, so the coefficient matrix is singular and its determinant is zero. The system is then either inconsistent, if some row reduces to all zeros with a nonzero constant, or underdetermined with infinitely many solutions described by one or more free variables. This tool stops and reports the column where that happened rather than dividing by a near-zero pivot.
How much work does elimination take as the system grows?
The elimination stage needs on the order of n^3 arithmetic operations for an n by n system, so doubling the number of unknowns multiplies the work by roughly eight. For hand calculation that means 2x2 and 3x3 systems are comfortable and anything larger is tedious. It also explains why large sparse systems in engineering are usually solved with iterative methods such as conjugate gradient rather than with direct elimination.
What is an ill-conditioned system and how would I spot one?
An ill-conditioned system is one where the equations are nearly linearly dependent, so the lines or planes they define intersect at a very shallow angle. A tiny change in a coefficient or a constant then swings the solution a long way. Warning signs are pivots that shrink dramatically as elimination proceeds and a determinant close to zero relative to the size of the entries. Partial pivoting limits rounding damage but cannot cure genuine ill-conditioning.
How does Gaussian elimination relate to LU decomposition and the determinant?
The multipliers used to zero out each column can be stored instead of discarded, and they assemble into a lower triangular matrix L while the reduced matrix is the upper triangular U, giving A = LU. That factorisation lets you solve the same coefficient matrix against many different right-hand sides cheaply. The determinant also falls out for free: it is the product of the pivots, with the sign flipped once for every row swap performed during pivoting.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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