Example data table
This sample 3×3 matrix is invertible and commonly used for practice.
| Sample A | ||
|---|---|---|
| 2 | 1 | 1 |
| 1 | 3 | 2 |
| 1 | 0 | 0 |
| Expected A−1 (rounded) | ||
|---|---|---|
| 0.0000 | 0.0000 | 1.0000 |
| -2.0000 | 1.0000 | 3.0000 |
| 3.0000 | -1.0000 | -5.0000 |
Use “Load Example” to auto-fill this matrix into the form.
Formula used
The calculator finds A−1 by Gauss–Jordan elimination. It forms the augmented matrix [A | I], then applies elementary row operations until the left side becomes the identity matrix I. The right side then becomes the inverse.
- [A | I] → [I | A−1] using row swaps, scaling, and row addition.
- Partial pivoting chooses the largest available pivot to reduce numerical instability.
- The determinant is tracked from pivot values and row swaps (a swap flips its sign).
How to use this calculator
- Select the matrix size and your desired decimal precision.
- Enter all values of the square matrix A in the grid.
- Optionally enable step output to review row operations.
- Press Submit to compute A−1, determinant, and condition estimate.
- Use Download CSV/PDF to export results for notes or reports.
When an inverse exists
A square matrix is invertible only when it has full rank, meaning its rows and columns are independent. In practice, the determinant must be nonzero, so small pivots during elimination often signal trouble. This calculator reports the determinant alongside the inverse to confirm invertibility. If the determinant is exactly zero, the matrix is singular and no inverse exists. Near zero values may still cause unreliable results for reliable computations.
Meaning of the inverse
The inverse is the matrix that undoes a linear transformation. If A maps an input vector x to an output b, then the inverse maps b back to x by solving x = A inverse times b. In applications, this supports solving linear systems, changing coordinate bases, and reversing scaling and rotation effects. Because inversion amplifies noise when data is uncertain, interpretation should consider measurement error in many workflows.
How the inverse is computed
The calculator uses Gauss Jordan elimination on an augmented matrix made by placing the identity matrix beside A. Row operations transform A into the identity, while the same operations applied to the identity build the inverse. Partial pivoting selects the strongest pivot in each column to reduce rounding issues. Optional step output shows the evolving augmented table so you can audit the procedure with consistent scaling.
Accuracy and conditioning
Numerical stability depends on how sensitive the solution is to small input changes. A matrix with a large condition number can produce an inverse with large entries even when A looks ordinary. The tool estimates conditioning using the one norm of A and A inverse, giving a quick sense of sensitivity. If the estimate is high, increase precision, verify carefully, or prefer solving systems directly before making decisions.
Practical checks and exports
A simple verification is multiplying A by its computed inverse and checking whether the product is close to the identity matrix. Small differences are expected due to rounding, but large off diagonal values indicate instability or bad input. The downloadable CSV and PDF outputs help document results, share matrices, and keep audit trails for coursework or engineering notes. Always store the chosen precision with exports for future reference.
FAQs
What sizes can I invert here?
Choose sizes from 2×2 up to 6×6. Larger matrices increase rounding risk and take more operations, so review the determinant and conditioning estimate for confidence.
Why does it say the matrix is singular?
A singular matrix has no inverse because its determinant is zero and its rows are dependent. Try correcting input values, reducing duplicates, or using a different model that solves Ax=b without inversion.
What does the condition estimate tell me?
It approximates how much input error can be amplified in the inverse and in solved vectors. A large estimate suggests sensitivity; increase precision, validate with a product check, or avoid explicit inversion.
How many decimals should I use?
Use 4 to 6 decimals for typical coursework. For nearly singular matrices or very large values, increase decimals and compare results across precisions to detect instability.
Can I use fractions or negative numbers?
Yes. Enter decimals like 0.25 for 1/4 and negative values normally. If you start from fractions, convert consistently to avoid typing errors and keep units aligned.
How do I verify the inverse is correct?
Multiply A by the computed inverse and confirm the result is close to the identity matrix. Differences near zero are normal, but large off diagonal values indicate instability or incorrect inputs.