Enter matrices, pick size, and calculate inverses. Get determinants, steps, and solutions in clean tables. Download results, compare examples, and learn faster today now.
| A (3×3) | ||
|---|---|---|
| 4 | 7 | 2 |
| 3 | 6 | 1 |
| 2 | 5 | 3 |
| b |
|---|
| 1 |
| 0 |
| 4 |
| A⁻¹ (approx.) | ||
|---|---|---|
| 1.4444 | -1.2222 | -0.5556 |
| -0.7778 | 0.8889 | 0.2222 |
| 0.3333 | -0.6667 | 0.3333 |
| x = A⁻¹b (approx.) |
|---|
| -0.7778 |
| 0.1111 |
| 1.6667 |
An inverse summarizes how a square matrix transforms every vector. If A is invertible, each right‑hand side b yields a unique solution x, and A⁻¹ can be reused for many b values. This calculator focuses on 2×2 through 6×6 problems common in classes and small models. Always review det(A) and stability diagnostics, especially when inputs have very different magnitudes. Use it to explore linear transformations and double‑check worked examples quickly today.
The tool forms [A | I] and applies row operations until the left block becomes I, producing A⁻¹ on the right. Partial pivoting swaps the strongest pivot into place to avoid dividing by tiny values. Work grows roughly as O(n³), so small n remains fast. The pivot tolerance ε marks pivots as effectively zero, and optional steps list the operations performed. Fraction inputs are accepted, then converted to decimals for computation.
det(A) is a quick invertibility test: det(A)=0 means no inverse. The calculator estimates det(A) using elimination with row swaps, matching the pivot strategy used for inversion. For 2×2 and 3×3, an adjoint approach based on cofactors and det(A) is available for learning and manual verification. A determinant near zero often indicates a nearly singular matrix.
Sensitivity is approximated by cond∞(A)=‖A‖∞·‖A⁻¹‖∞. Values close to 1 suggest a well‑conditioned matrix, while large values imply amplified rounding effects. The check max |I − A·A⁻¹| verifies that the computed inverse reproduces the identity matrix. If error is high, rescale inputs, tighten ε carefully, or use simpler fractions. Display precision affects tables only. Aim for errors near 0 within tolerance.
Enable Ax=b to compute x=A⁻¹b after inversion and compare solutions across different b vectors. For a single solve, direct elimination is usually preferred, but the inverse method is useful for instruction and repeated solves. CSV exports suit spreadsheets and plotting, while the PDF summary suits assignments and reviews. Keep reports with inputs, rounding, and diagnostics for traceability. Export both formats to keep a clear record.
It supports square matrices from 2×2 up to 6×6. Select a size, and the input grids update for matrix A and vector b.
An inverse does not exist when det(A)=0, meaning A is singular. If pivots fall below the tolerance ε, the tool reports the matrix as singular or nearly singular.
Adjoint and cofactor formulas grow expensive and can overflow as size increases. Gauss–Jordan scales more predictably, uses pivoting for better stability, and fits typical 4×4 to 6×6 classroom problems.
It estimates how sensitive results are to small input changes. A low value suggests stable inversion, while a high value means rounding or measurement noise can noticeably change A⁻¹ and x.
Keep ε small, such as 1e-12 for well-scaled inputs. If your matrix entries are very small or very large, rescale the matrix first; increasing ε too much may incorrectly reject valid pivots.
Yes. Exports include your matrix A, vector b, det(A), cond∞(A), the inverse A⁻¹, and the optional solution x. Use them to document inputs, rounding, and computed outputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.