Condition Number Calculator - Matrix Conditioning & Stability

The condition number of an invertible matrix A is defined as κ(A) = ‖A‖ · ‖A⁻¹‖, where ‖·‖ denotes any compatible matrix norm. It measures how much the relative error in the solution x of a linear system A·x = b can be amplified by a relative error in b. Intuitively, a matrix with a small condition number is well-conditioned: small input errors produce small output errors. A matrix with a large condition number is ill-conditioned: even microscopic floating-point round-off in the data can produce wildly inaccurate solutions. This calculator supports 2 × 2 and 3 × 3 matrices and three of the most widely used matrix norms. The 1-norm is the maximum absolute column sum, ‖A‖₁ = max_j Σᵢ |aᵢⱼ|. The infinity-norm is the maximum absolute row sum, ‖A‖∞ = max_i Σⱼ |aᵢⱼ|. The Frobenius norm is the square root of the sum of the squares of all entries, ‖A‖_F = √(Σᵢⱼ |aᵢⱼ|²), and is the matrix analogue of the Euclidean vector norm. For 2 × 2 matrices the inverse is computed analytically as A⁻¹ = (1/det A) · [[d, −b], [−c, a]]. For 3 × 3 matrices the cofactor (adjugate) formula is used. Condition numbers are central to numerical linear algebra. When you solve a linear system on a computer using LU decomposition or Gaussian elimination, the relative error in the computed solution is bounded above by roughly κ(A) · ε, where ε is the machine precision (about 10⁻¹⁶ in IEEE-754 double precision). A condition number of 10⁶ therefore means you can lose up to six digits of accuracy from round-off alone. As a rough rule of thumb, matrices with κ < 100 are considered well-conditioned, those with κ between 100 and 1000 are moderately conditioned, and anything above 10³ is ill-conditioned and should be approached with care. A few important caveats. The condition number depends on the choice of norm, so values computed with different norms are not directly comparable, though they are usually within a small constant factor of each other. The 2-norm (spectral norm) condition number, defined via singular values, is the most theoretically natural choice but is more expensive to compute and is not offered here. A singular matrix has determinant exactly zero, no inverse, and an infinite condition number; the calculator detects this case explicitly. Use this tool whenever you need to check whether a small linear system is safe to invert numerically, when teaching introductory numerical analysis, or as a quick sanity check before solving a system in a larger simulation or machine-learning pipeline.