Matrix Inverse Calculator
Invert a square matrix with Gauss-Jordan elimination under partial pivoting.
Description
Invert a non-singular square matrix with Gauss-Jordan elimination and partial pivoting.
The inverse of a square matrix A is the matrix A⁻¹ for which A·A⁻¹ equals the identity matrix. Inverses are used to solve linear systems, reverse linear transformations, and verify matrix algebra, although direct factorization is often preferable in numerical software.
When to use Matrix Inverse Calculator
- Check a small inverse calculation
- Reverse an invertible linear transformation
- Verify that a product reproduces the identity matrix
How the calculation works
Gauss-Jordan elimination augments the input with an identity matrix. Partial pivoting swaps in the largest available pivot, each pivot row is scaled to one, and the pivot column is eliminated from every other row. The transformed identity side is the inverse.
A · A⁻¹ = I Interpreting the result
Multiply the returned matrix by the input to check that the result is approximately the identity. Large inverse entries can signal that the input is close to singular and that small data errors may be strongly amplified.
Important limitations
- Only non-empty square matrices can be inverted.
- A singular matrix is rejected because no unique inverse exists.
- Entries are rounded to twelve significant digits, and ill-conditioned matrices can lose substantially more accuracy.