Gauss-Jordan Elimination Calculator
Solve systems of equations, reduce matrices to RREF, find matrix inverses, and see every Gauss-Jordan row operation step by step with pivot highlights and exact fraction results.
Background
Gauss-Jordan elimination is a row-reduction method that transforms a matrix into reduced row echelon form. It is used to solve linear systems, identify free variables, test consistency, find rank, and compute inverse matrices.
How to use this calculator
- Choose solve system, matrix to RREF, or inverse matrix mode.
- Select the matrix size, enter each value, or paste a full matrix into the paste box.
- Use fractions such as 3/4 when you want exact arithmetic.
- Click Calculate to see REF vs RREF, determinant when available, row-operation counts, rank, pivot columns, and solution interpretation.
- Use quick examples to practice unique solutions, infinite solutions, inconsistent systems, and inverse matrices.
How Gauss-Jordan elimination works
- Find a nonzero pivot in the current column.
- Swap rows if needed to move the pivot into position.
- Scale the pivot row so the pivot becomes 1.
- Eliminate every other entry in that pivot column.
- Repeat until the matrix is in reduced row echelon form.
Formula & Row Operations Used
Swap rows: Ri ↔ Rj
Scale a row: Ri ← kRi, where k ≠ 0
Add a multiple of another row: Ri ← Ri + kRj
Reduced row echelon form: A → RREF(A)
Inverse setup: [A | I] → [I | A−1]
Example Problems & Step-by-Step Solutions
Example 1: Solve a 2×2 system
For a system such as x + y = 5 and 2x − y = 1, write the augmented matrix [A | b].
Gauss-Jordan elimination reduces the matrix until the left side becomes the identity matrix.
The final constants column gives the solution values for the variables.
Example 2: Identify infinite solutions
If a system has fewer pivot columns than variables and no contradiction row, at least one variable is free.
The calculator names the free variables and writes the solution in terms of parameters.
Example 3: Find an inverse matrix
To find A−1, the calculator starts with [A | I].
If the left side can be reduced to the identity matrix, the right side becomes the inverse.
If a pivot is missing, the matrix is singular and has no inverse.
Common mistakes to avoid
- Do not stop at row echelon form if the problem asks for reduced row echelon form.
- Do not forget to divide the whole row when making a pivot equal to 1.
- Do not change only one side of an augmented matrix. Row operations apply to the entire row.
- Do not assume a row of zeros means no solution. A contradiction row such as 0 = 5 means no solution.
- Do not ignore free variables when a pivot column is missing.
Frequently Asked Questions
What is Gauss-Jordan elimination?
Gauss-Jordan elimination is a row-reduction method that transforms a matrix into reduced row echelon form by creating pivots and eliminating entries above and below each pivot.
What is the difference between Gaussian elimination and Gauss-Jordan elimination?
Gaussian elimination usually stops at row echelon form and then uses back substitution. Gauss-Jordan elimination continues until the matrix reaches reduced row echelon form.
Can this calculator solve systems with infinite solutions?
Yes. If the system is consistent but has free variables, the calculator identifies the free variables and writes the solution using parameters.
Can this calculator find inverse matrices?
Yes. In inverse mode, the calculator row-reduces [A | I]. If the left side becomes the identity matrix, the right side is the inverse.
Why are fractions useful in Gauss-Jordan elimination?
Fractions avoid rounding errors. This calculator keeps exact fraction values during row operations so students can follow the algebra clearly.
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