🔗 Systems Calculators

Gaussian Elimination: The Universal Method

Gaussian elimination reduces a system of linear equations to row echelon form using three elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. These operations preserve the solution set. Once in row echelon form, back-substitution solves for variables from bottom to top. Gauss-Jordan elimination goes further, reducing to reduced row echelon form (RREF) where each pivot is 1 and all other entries in the pivot column are 0 — this directly reveals the solution without back-substitution. For a 3×3 system, Gaussian elimination requires approximately n³/3 arithmetic operations, making it the standard algorithm for numerical linear algebra even in modern software.

Cramer's Rule

Cramer's rule solves a system Ax = b by expressing each variable as a ratio of determinants: xᵢ = det(Aᵢ) ÷ det(A), where Aᵢ is the matrix A with its i-th column replaced by the vector b. For a 2×2 system ax + by = e and cx + dy = f: x = (ed − bf)/(ad − bc) and y = (af − ce)/(ad − bc). Cramer's rule is elegant and produces explicit closed-form solutions, making it useful for symbolic algebra and understanding how solutions depend on parameters. However, for large systems it is computationally expensive (O(n!) in the naive implementation), so Gaussian elimination is preferred numerically. Cramer's rule requires det(A) ≠ 0; if the determinant is zero, the system either has no solution or infinitely many.

The Augmented Matrix and RREF

An augmented matrix [A|b] combines the coefficient matrix A with the right-hand-side vector b into a single matrix, streamlining the row reduction process. Row-reducing [A|b] to RREF directly reads off the solution — or reveals inconsistency (a row [0 0 ... 0 | c] with c ≠ 0 means no solution) or dependency (a row of all zeros means a free variable and infinitely many solutions). The RREF is unique for any matrix, making it a canonical form for analysing the solution structure. For homogeneous systems (b = 0), RREF reveals the null space of A — the set of all x satisfying Ax = 0. The null space has dimension n − rank(A) by the rank-nullity theorem.

Matrix Equation AX = B

The matrix equation AX = B generalises simultaneous equations to multiple right-hand sides at once. When A is invertible, the unique solution is X = A⁻¹B. This approach is efficient when solving many systems with the same coefficient matrix A but different b vectors — compute A⁻¹ once, then multiply by each b column. In practice, computing A⁻¹ explicitly is numerically unstable; most software instead computes the LU factorisation of A (A = LU where L is lower triangular and U is upper triangular) and solves LUX = B via forward and back substitution. The LU decomposition is used in circuit simulation (SPICE), finite element analysis, and computational fluid dynamics wherever large sparse linear systems must be solved repeatedly.