Matrix Inverse Calculator

Paste a square matrix to compute its inverse using Gauss-Jordan elimination in your browser.

Stable numericsSubstitute / reconstructIndependent checkLocal calculation

Square matrix (rows on new lines)

Inverse calculatedGauss-Jordan elimination

Identity residual & conditioning

Recompute the inverse and quantify how closely A×A⁻¹ matches identity instead of trusting output formatting alone.

Verification evidence

Residual tolerance

Norm

Preview decimals

Row	Largest identity error

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Audit uses the visible inputs plus the assumptions shown here.

Algebra round-trip & degeneracy audit

Reconstruct the same mathematical object through an independent equivalent form, then measure the residual and classify degenerate cases explicitly. The tolerance is a numerical review threshold, not proof for arbitrary symbolic expressions.

Residual toleranceRound-trip scale factorEvaluation probe x

Invariant / round trip	Result	Residual / classification

Change the calculator inputs above, then refresh this algebra audit.

Equivalent-form checks reduce transcription risk but do not replace domain restrictions or symbolic proof.

Algebra round-trip cluster

Solve once, then verify the same object through another representation

Independent round trips and invariants expose sign, scaling, reconstruction and degeneracy mistakes that a single formatted answer can hide.

Quadratics Matrices Systems

All 28 algebra tools

Matrix InverseA↔A⁻¹ identity round trips MidpointEndpoint reflection round trip Linear EquationScale invariance + degeneracy classes DeterminantTranspose/row-operation invariants Quadratic VertexStandard ↔ vertex coefficient round trip Slope InterceptPoints ↔ line-form equivalence Polynomial DivisionQuotient/remainder reconstruction Standard FormSlope-intercept ↔ standard round trip

Practical guide and verification

An inverse may not exist

A square matrix has an inverse only when it is nonsingular. If the determinant is zero, or the rows/columns are linearly dependent, there is no ordinary inverse to compute.

Near-singular matrices

A matrix can be technically invertible yet numerically ill-conditioned. Small changes in the inputs may then produce large changes in the inverse, so a long decimal output can look more trustworthy than the data justify.

Common mistake

Do not use matrix inversion as an automatic replacement for solving a linear system. Numerical methods that solve Ax=b directly are often more stable and efficient than explicitly forming A⁻¹.

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Inverse identity verification

Gauss–Jordan output is independently checked by multiplying the input matrix by the inverse and measuring the identity residual.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.