How to use the Gauss-Jordan Method Calculator
- 1. Choose a taskSolve Ax = b with an augmented matrix, or find A⁻¹ from a square matrix.
- 2. Enter the matrixOne row per line. Fractions such as 1/2 and decimals are accepted.
- 3. Follow the stepsEach column shows the pivot scaling and the rows cleared above and below it.
How Gauss-Jordan elimination works
Gauss-Jordan goes one step further than Gauss elimination. For each column it divides the pivot row by the pivot so the pivot becomes 1, then makes every other entry in that column zero, above and below. After the last column, the left part is the identity matrix and the right part is the answer.
To find the inverse, write [A | I] and apply the same operations. When A turns into I, the right half has turned into A⁻¹. If a column has no non-zero pivot, A is singular and has no inverse.
Worked example
Solve x + y + z = 9, 2x − 3y + 4z = 13, 3x + 4y + 5z = 40. Column 1: R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁ give [0, −5, 2 | −5] and [0, 1, 2 | 13]. Column 2: R₂ → R₂ ÷ (−5), then clear R₁ and R₃, giving R₃ = [0, 0, 12/5 | 12]. Column 3: R₃ → R₃ ÷ (12/5), then clear above.
The final matrix is [I | (1, 3, 5)], so x = 1, y = 3, z = 5. For the inverse of [[2,1,1],[1,2,1],[1,1,2]] the tool gives (1/4)[[3,−1,−1],[−1,3,−1],[−1,−1,3]].
When to use it
Gauss-Jordan needs about 50% more arithmetic than Gauss elimination for one system, but it is the standard way to find an inverse by hand. For a triangular factorisation that can be reused for many right-hand sides, use LU decomposition.
Frequently asked questions
How is the inverse found?
The tool row-reduces [A | I]. When the left half becomes I, the right half is A⁻¹. You can check by multiplying A by the result.
What if the matrix is singular?
Some column will have no pivot. The tool reports that rank(A) is less than n, so A⁻¹ does not exist.
Does it handle no solution or infinite solutions?
Yes. A row like [0 … 0 | c] with c ≠ 0 means no solution; fewer pivots than unknowns means infinitely many.
Is the answer exact?
Yes. All work is in exact fractions unless you choose decimals for display.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.