Matrix Inverse Calculator
The inverse of a square matrix by Gauss-Jordan elimination, with the identity check that proves it.
Results
What this tool does
The inverse of a matrix undoes what the matrix does, and it is how a system of equations gets solved in one step instead of many. Not every matrix has one: if the determinant is zero the matrix flattens space and nothing can unflatten it. This page reduces your matrix alongside the identity until the roles swap, which is the method taught and the one that works at any size. It then multiplies the two back together and shows you the result, because an inverse you cannot check is an inverse you should not use.
Formula
[A | I] → [I | A⁻¹] (Gauss-Jordan) · A × A⁻¹ = I
Variables
| Symbol | Meaning | Unit |
|---|---|---|
a | The matrix, one row per line | — |
IV | The inverse | — |
OK | Does it have an inverse? | — |
DT | Determinant | — |
CK | The matrix times its inverse | — |
CE | How far that is from the identity | — |
SZ | Size | — |
BG | Largest entry | — |
OG | The matrix as it was read | — |
Worked example
- The matrix, one row per line4, 7 2, 6
- The inverse 0.600000 -0.700000 -0.200000 0.400000
- Does it have an inverse?Yes
- Determinant10.00000000
- The matrix times its inverse1 0 0 1
- How far that is from the identity2.22045e-16
- Size2 x 2
- Largest entry0.70000000
- The matrix as it was read4 7 2 6
Frequently asked questions
Why does the page show A times its inverse?
Because that product must be the identity matrix — ones down the diagonal and zeros everywhere else — and seeing it is the only way to know the answer is right without redoing the work. Inverting a matrix by hand is error-prone and the mistakes are silent; multiplying back is the standard check and takes seconds. The largest deviation from the identity is shown as a number; anything beyond about 1e-10 means the matrix is close to singular and the inverse should not be trusted.
When does a matrix have no inverse?
When its determinant is zero, which means one of its rows can be built from the others — the matrix squashes space flat and there is no way to unsquash it. In practical terms it means the system of equations it represents either has no solution or has infinitely many, never exactly one. A determinant that is not zero but very small is nearly as bad: the inverse exists on paper but its entries are enormous, and small errors in the input become large errors in the output.