Matrix inverse calculator
Invert a square matrix and follow every row operation — by Gauss-Jordan elimination on [A | I], the adjugate formula or Montante's method. Exact fractions, not rounded decimals.
Enter your matrix, open the Advanced tab and press A⁻¹. If the matrix is singular you get a clear message instead of a wrong answer. The step-by-step panel shows the intermediate values as exact fractions — 3/5 rather than 0.6000000001 — because it runs on rational arithmetic rather than floating point.
When an inverse exists
Only for square matrices, and only when the determinant is non-zero. The inverse is the matrix that undoes the original: A·A⁻¹ = I. There is no such thing as an approximate inverse of a singular matrix — if the determinant is zero, information was destroyed and cannot be recovered.
Worked example
Gauss-Jordan writes the identity beside the matrix and reduces the left half to I:
| 4 | 7 | 1 | 0 |
| 2 | 6 | 0 | 1 |
| 1 | 0 | 3/5 | −7/10 |
| 0 | 1 | −1/5 | 2/5 |
Whatever the right half became is the inverse. Always verify by multiplying: A·A⁻¹ should give exactly the identity, and the A × B button does that check for you in two clicks.
Common mistakes
- Inverting the entries one by one. The inverse is not the matrix of reciprocals —
1/a(i,j)has nothing to do with it. - Getting the order wrong.
(A·B)⁻¹ = B⁻¹·A⁻¹, notA⁻¹·B⁻¹. - Solving systems through the inverse. For
Ax = b, row-reducing[A | b]is fewer operations and numerically more accurate. - Using the adjugate on big matrices. It needs
n²determinants of sizen−1. Past 3×3 it is far slower than elimination.