System of linear equations calculator

Gauss-Jordan elimination turns the system into an equivalent one — one with the same solutions — whose answer can be read off directly. You work on the augmented matrix, coefficients plus constants, using just three legal moves: swap two rows, multiply a row by a non-zero number, add a multiple of one row to another. At the end every pivot is 1 and is the only non-zero entry in its column.

The classification follows the Rouché-Capelli theorem, which compares two ranks: that of the coefficient matrix and that of the augmented matrix. If the second is greater than the first, elimination has produced a row of the form 0 = c with c non-zero — a contradiction — and the system is inconsistent. If the two ranks match there are solutions: exactly one when the rank equals the number of unknowns, infinitely many when it is lower.

In the indeterminate case the unknowns split into two groups. Those matching a pivot column are determined by the others; the rest are free and can take any value. The general solution is then written as a particular solution plus a linear combination of the free directions, one for each free unknown: it is the same structure that in geometry describes a line or a plane of solutions.

The calculation uses partial pivoting: at each step the pivot chosen is the entry of largest magnitude in the column, not simply the first non-zero one. This is not a matter of efficiency but of reliability: a very small pivot would amplify every rounding error made afterwards, and that is how a perfectly solvable system ends up looking inconsistent.