Matrix Determinant + Inverse (2×2 / 3×3)
Determinant and inverse for 2×2 or 3×3 matrices. Inverse exists iff det ≠ 0.
Result
- Matrix[[4, 7], [2, 6]]
- det = ae − bd10.000000
- Inverse [0,0] = e/det0.600000
- Inverse [0,1] = −b/det-0.700000
- Inverse [1,0] = −d/det-0.200000
- Inverse [1,1] = a/det0.400000
Step-by-step
- 2×2 det = ae − bd = 4·6 − 7·2 = 10.0000.
- 2×2 inverse = (1/det) · [[e, −b], [−d, a]] (swap diagonal, negate off-diagonal, divide by det).
How to use this calculator
- Pick 2×2 or 3×3 from the dropdown.
- Enter entries row-by-row. For 2×2, only a/b/d/e matter.
- Read the determinant; the inverse appears below if det ≠ 0.
About this calculator
Two operations central to linear algebra. The determinant det(M) is a scalar that measures volume scaling under the linear transformation M (and orientation: negative det = reflection). For a square matrix M, the inverse M⁻¹ exists if and only if det(M) ≠ 0 (the matrix is non-singular). 2×2 inverse: swap diagonal entries, negate off-diagonal, divide by det. 3×3 inverse via the adjugate (= transpose of the cofactor matrix) divided by the determinant: M⁻¹ = adj(M)/det. For larger matrices the formula extends but is computationally expensive; Gaussian elimination is preferred from 4×4 upward. Singular matrices arise when rows (or columns) are linearly dependent — geometrically, the transformation collapses the space onto a lower-dimensional subspace.