Matrix Inverse Calculator
The Matrix Equivalent of Division
Ordinary numbers have reciprocals; square matrices have inverses, and they play the same role — a matrix multiplied by its own inverse yields the identity matrix, the matrix world's version of the number 1. Not every matrix has one: a matrix is invertible only when its determinant is non-zero.
The Formulas
2×2: A-1 = (1/det(A)) × [[d, −b], [−c, a]]
3×3: A-1 = (1/det(A)) × adj(A)
3×3: A-1 = (1/det(A)) × adj(A)
Worked Example
For matrix A = [4, 7; 2, 6]:
| Step | Calculation | Result |
|---|---|---|
| det(A) | (4×6) − (7×2) | 10 |
| Inverse | (1/10) × [[6, −7], [−2, 4]] | [[0.6, −0.7], [−0.2, 0.4]] |
Where This Matters
- Solving linear systems as Ax = b — multiplying both sides by A-1 isolates x directly, an approach used throughout engineering and computer graphics.
- Coordinate transformations — reversing a rotation, scaling, or shear transformation applies the inverse of the original transformation matrix.
- Statistics and regression — least-squares regression coefficients are computed using a matrix inverse (or a numerically stable equivalent).
Note: If the determinant is zero, the matrix is singular and has no inverse — this calculator will flag that case rather than divide by zero.
How to Use This Calculator
- Choose the matrix size: 2 x 2 Matrix or 3 x 3 Matrix.
- Enter each cell value (row1 col1, row1 col2, and so on).
- Select Calculate to get the inverse matrix.
Related Calculations
Check invertibility first with the Matrix Determinant Calculator, or multiply matrices together with the Matrix Multiplication Calculator.