Matrix Determinant Calculator 3×3 (with Inverse Matrix)
By Santaji GadepatilinUpdated November 10, 2026
The determinant of a square matrix is a single number that tells you whether the matrix can be inverted, how it scales areas and volumes, and whether a system of linear equations has a unique solution. This matrix determinant calculator expands a 3×3 matrix by cofactors, gives its trace and, when the determinant is not zero, the full inverse matrix from the adjugate.
How to use the matrix determinant calculator
Enter the nine elements of the 3×3 matrix row by row — a₁₂ is row 1, column 2. The calculator gives the determinant, the trace and the nine elements of the inverse matrix. If the determinant is zero the matrix is singular and the inverse elements show as infinite or undefined. Fractions such as 1/3 can be typed as decimals.
Determinant and inverse formulas
det A = a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31)
Cij = (−1)i+j Mij · A⁻¹ = adj(A) ÷ det A = CT ÷ det A
Mij is the minor — the determinant of the 2×2 matrix left after deleting row i and column j. For a 2×2 matrix the determinant is ad − bc and the inverse is [d −b ; −c a] ÷ (ad − bc).
Worked example
A = [2 1 1 ; 1 3 2 ; 1 0 0] (rows separated by semicolons):
- det A = 2 × (3 × 0 − 2 × 0) − 1 × (1 × 0 − 2 × 1) + 1 × (1 × 0 − 3 × 1) = 0 + 2 − 3 = −1
- Trace = 2 + 3 + 0 = 5
- Cofactor matrix C = [0 2 −3 ; 0 −1 1 ; −1 −3 5]
- Inverse A⁻¹ = Cᵀ ÷ (−1) = [0 0 1 ; −2 1 3 ; 3 −1 −5]
- Check: A × A⁻¹ gives the identity matrix
Determinants of common matrices
| Matrix | Elements (rows) | Determinant | Inverse |
|---|---|---|---|
| Identity | 1 0 0 ; 0 1 0 ; 0 0 1 | 1 | invertible |
| Diagonal 2, 3, 4 | 2 0 0 ; 0 3 0 ; 0 0 4 | 24 | invertible |
| Rotation 90° about z | 0 −1 0 ; 1 0 0 ; 0 0 1 | 1 | invertible |
| Upper triangular | 1 2 3 ; 0 4 5 ; 0 0 6 | 24 | invertible |
| Rows 1 2 3 / 4 5 6 / 7 8 9 | 1 2 3 ; 4 5 6 ; 7 8 9 | 0 | singular — no inverse |
| This page’s example | 2 1 1 ; 1 3 2 ; 1 0 0 | −1 | invertible |
Properties of determinants
- Swapping two rows changes the sign of the determinant; multiplying a row by k multiplies it by k.
- Adding a multiple of one row to another leaves the determinant unchanged — the basis of Gaussian elimination.
- det(AB) = det A × det B, det(Aᵀ) = det A and det(A⁻¹) = 1 ÷ det A.
- The determinant of a triangular matrix is the product of its diagonal elements.
- |det A| is the volume of the parallelepiped spanned by the rows; its sign tells you whether the orientation is flipped.
Practical notes
- A determinant that is tiny compared with the size of the elements means the matrix is ill-conditioned: small changes in the data cause large changes in the inverse or in the solution of the equations.
- To solve A x = b for three unknowns, use the system of 3 equations calculator — it applies Cramer’s rule, which is det-based too.
- In engineering, 3×3 determinants appear in vector cross products, rotation matrices, stress tensors and the Clarke and Park transformations.
Frequently asked questions
How do you find the determinant of a 3×3 matrix?
Expand along the first row, det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁). Each bracket is the determinant of the 2×2 minor left after deleting that element's row and column, and the signs alternate +, −, +.
How do you find the inverse of a 3×3 matrix?
Replace every element by its cofactor, transpose the result to get the adjugate, and divide by the determinant, A⁻¹ = adj(A) ÷ det(A). Check it by multiplying, A × A⁻¹ = I.
What does a determinant of zero mean?
The matrix is singular — its rows (and columns) are linearly dependent, it squashes space into a plane or line, and it has no inverse. A system of equations with that coefficient matrix has either no solution or infinitely many.
How do I use this for a 2×2 matrix?
Enter the 2×2 matrix in the top-left corner, set a₃₃ = 1 and the other entries of row 3 and column 3 to 0. The determinant is then ad − bc and the top-left block of the inverse is the 2×2 inverse.