Matrix Input
Max 12×12
For Fraction, limit denominator for readable rational forms.
Tip: Paste CSV or tab-separated values with the "Paste CSV" button.
Results
Inverse Matrix A−1
Validation
Example Data Table
Example 3×3 matrix. Click to load into the calculator.
| 2 | -1 | 0 |
| -1 | 2 | -1 |
| 0 | -1 | 2 |
Formula Used
We form the augmented matrix [A | I] and apply elementary row operations to reach [I | A−1]. Operations include row swaps, scaling a row by a nonzero scalar, and adding a multiple of one row to another.
- Partial pivoting selects the largest available pivot in each column to improve numerical stability.
- Invertibility requires a nonzero determinant. Near-zero pivots signal ill-conditioning or singularity.
- Complexity is O(n³). Residuals and a crude condition estimate are reported after inversion.
- Fraction display uses rational approximation for readability, not symbolic exact arithmetic.
How to Use
- Set the dimension (up to 12) and preferred precision.
- Enter values directly, paste CSV, or load the example.
- Choose pivoting strategy. Partial is recommended.
- Pick a display: Auto, Fixed, Exponential, or Fraction.
- For Fraction, adjust Max Denominator to simplify results.
- Click Compute Inverse. Review steps and validation.
- Download the inverse as CSV or a PDF report.
FAQs
A square matrix A is invertible if and only if det(A) ≠ 0. Zero or tiny pivots indicate singularity or severe ill-conditioning.
Pivoting reduces round-off error by avoiding divisions by very small numbers, leading to a more stable inverse.
Use 6–8 digits for general work. Increase precision for ill-conditioned problems, but note the limits of floating-point arithmetic.
Each step records the pivot, any row swaps, row scaling to make a 1, and eliminations above/below to zero out the column.
The process halts with an error. Consider revising the matrix or using a pseudoinverse approach specific to your application.
Yes, up to 12×12 here for usability. Larger sizes are possible but become slower and harder to audit visually.