Matrix Calculator
Compute the determinant and inverse of a 2×2 or 3×3 matrix.
Input Data
Results
At a glance:The determinant is a scalar telling whether a matrix is invertible; the inverse A⁻¹ satisfies A·A⁻¹ = I. 2×2 has a simple formula; 3×3 uses cofactors. det=0 means no inverse. The determinant also measures how a linear transform scales area/volume.
Formula
2×2: det = ad − bc; A⁻¹ = (1/det)·[[d,−b],[−c,a]].
3×3: cofactor expansion along the first row.
det=0 ⇒ no inverse (singular).
How to Use
- Pick a size (2×2 or 3×3).
- Fill the elements aᵢⱼ.
- Read determinant and inverse.
- If det=0 the note explains there is no inverse.
Determinant & inverse notes
| Item | 2×2 | 3×3 |
|---|---|---|
| Determinant | ad−bc | cofactor expansion |
| Inverse exists | det≠0 | det≠0 |
| A⁻¹ form | (1/det)[[d,−b],[−c,a]] | adj(A)/det |
det≠0 is necessary and sufficient for an inverse to exist.
Case Studies
2×2: [[4,7],[2,6]]
det = 4·6 − 7·2 = 10.
A⁻¹ = (1/10)·[[6,−7],[−2,4]] = [[0.6,−0.7],[−0.2,0.4]].
2×2 singular: [[1,2],[2,4]]
det = 1·4 − 2·2 = 0 → no inverse (columns proportional).
3×3 diagonal: diag(2,3,4)
det = 2·3·4 = 24.
A⁻¹ = diag(0.5, 0.3333, 0.25).
3×3 identity I
det = 1; A⁻¹ = I (identity is its own inverse).
Rotation [[0,−1],[1,0]]
det = 0·0 − (−1)·1 = 1.
Inverse equals transpose [[0,1],[−1,0]] (orthogonal matrix).
Geometric meaning
The 2×2 det is the signed area of the parallelogram spanned by its columns.
The 3×3 det is the signed volume of the parallelepiped.
FAQ
Non-square matrices?
This tool handles only square 2×2 and 3×3 matrices; non-square has no determinant (a pseudoinverse exists but is out of scope).
Why no inverse sometimes?
A matrix with determinant 0 is singular — its columns are linearly dependent, so no inverse exists.
What does the determinant mean?
det≠0 means invertible; its absolute value is the area/volume scaling of the linear map; the sign tells whether orientation flips.
2×2 inverse formula?
A⁻¹ = (1/det)·[[d,−b],[−c,a]]: swap the diagonal, negate the off-diagonal, divide by det.
How to invert 3×3?
Compute each cofactor to get the adjugate adj(A), transpose it, then divide by det: A⁻¹ = adj(A)/det.
Transpose vs inverse?
Transpose Aᵀ swaps rows and columns; inverse A⁻¹ satisfies A·A⁻¹=I. Only orthogonal matrices have A⁻¹=Aᵀ.
Relation to solving equations?
For Ax=b with square A, if invertible x=A⁻¹b; also solvable by Cramer's rule. det=0 means no unique solution.
What is the inverse used for?
Solving linear systems, coordinate changes, cryptography and circuit analysis; also inverse transforms in computer graphics.
Related Tools
References
Content review: Calculatorism Science Team. Determinant and inverse formulas verified. Results are for reference only.