Linear System Solver (2×2 / 3×3) – Tutorial

What Is a Linear System of Equations? (Ax=b)

A system of linear equations requires multiple linear equations to be satisfied simultaneously. Matrix form:

A·x = b

• A: Coefficient matrix (2×2 or 3×3)
• x: Vector of unknowns (x, y, z)
• b: Constants vector

Solution Types

• Unique solution: System is fully determined. Usually occurs when det(A) ≠ 0.
• Infinite solutions: Equations are dependent, free variables exist.
• No solution: Contradiction exists: e.g. 0 = 5.

How Does This Tool Solve? (Gauss–Jordan / RREF)

The Gauss–Jordan method takes the augmented matrix [A|b] and transforms it into Reduced Row Echelon Form (RREF) using row operations.

Allowed row operations

• Swap two rows
• Multiply a row by a nonzero scalar
• Add/subtract a multiple of one row to another

Pivot logic

Selecting the largest (in absolute value) element in each column as the pivot improves numerical stability (partial pivoting). The pivot row is then normalized to 1, and all other rows in that pivot column are zeroed out.

Decision via Rank (r(A), r([A|b]))

• r(A) = r([A|b]) = number of variables → unique solution
• r(A) = r([A|b]) < number of variables → infinite solutions
• r(A) < r([A|b]) → no solution

What Is the Determinant For?

For 2×2 and 3×3 systems, the determinant helps quickly assess whether a unique solution exists:

• If det(A) ≠ 0, there is usually a unique solution.
• If det(A) = 0, there may be infinite solutions or no solution (check rank).

Ill-Conditioning Warning

If pivots are very small or det(A) is near zero, the system may be ill-conditioned. In this case, a small data error can drastically change the solution. This tool performs a simple "pivot ratio" check to flag potential ill-conditioning.

This tool is numerical; it provides practical solutions and reporting rather than rigorous mathematical proof.