About the Determinant Calculator
This determinant calculator finds det(A) for any square matrix from 2×2 up to 6×6. Type the matrix with one row per line (or rows separated by semicolons), using whole numbers, decimals or fractions such as 1/2. The answer is computed in exact rational arithmetic, so there is no rounding error — a determinant that should be 0 really comes out as 0.
To show the working, the calculator expands along the row or column with the most zeros (Laplace or cofactor expansion): it lists each entry, its sign from the checkerboard pattern, the determinant of its minor and the resulting term, then adds the terms. For a 2×2 matrix this reduces to the familiar ad − bc.
Use it to check linear algebra homework, test whether a system of equations has a unique solution, decide if a matrix is invertible, or find the area or volume scale factor of a linear transformation. A non-zero determinant means the matrix is invertible; zero means its rows are linearly dependent.
With the default inputs, the determinant is 49. Change any value above to recalculate instantly.
How to use the determinant calculator
- 1Type the matrix with one row per line and spaces or commas between entries.
- 2Use fractions like 2/3 or decimals if needed — the result stays exact.
- 3Read the determinant and its exact value.
- 4Check the cofactor table to follow each term of the expansion.
- 5If the determinant is 0, the matrix has no inverse.
Formula and method
The Laplace (cofactor) expansion writes the determinant of an n×n matrix as a sum over one row or column: each entry aᵢⱼ is multiplied by its minor Mᵢⱼ (the determinant of the matrix left after deleting row i and column j) and by the checkerboard sign (−1)^(i+j). Choosing the row or column with the most zeros skips the most work.
Each minor is itself evaluated exactly with fraction-preserving Gaussian elimination: the matrix is reduced to upper-triangular form, the determinant is the product of the diagonal pivots, and every row swap flips the sign. Because all arithmetic uses exact fractions, the answer has no floating-point error.
- aᵢⱼ
- Entry in row i, column j
- Mᵢⱼ
- Minor: determinant with row i and column j removed
- (−1)^(i+j)
- Checkerboard sign (+ on the main diagonal)
- Cᵢⱼ
- Cofactor (−1)^(i+j)·Mᵢⱼ
Worked examples
3×3 determinant
Row 2 contains a zero, so the expansion runs along it: −2·(−3·5 − 1·4) + 0 − (−1)·(2·4 − (−3)·1) = 38 + 11 = 49.
2×2 matrix
For a 2×2 matrix the determinant is ad − bc = 4·6 − 7·2 = 24 − 14 = 10.
4×4 determinant
Column 2 has three zeros, so only the entry 1 in row 3 contributes: (−1)^(3+2)·1·M₃₂, where the 3×3 minor from rows 1, 2, 4 and columns 1, 3, 4 equals −30. The determinant is therefore 30.
Singular matrix with fractions
ad − bc = (1/2)·6 − 1·3 = 3 − 3 = 0. The second row is 6 times the first, so the matrix is singular.
Frequently asked questions
How do you find the determinant of a 3×3 matrix?+
Expand along the first row: a(ei − fh) − b(di − fg) + c(dh − eg) for the matrix with rows (a, b, c), (d, e, f), (g, h, i). The rule of Sarrus (diagonal products) gives the same answer, but only works for 3×3.
What does a determinant of zero mean?+
A zero determinant means the matrix is singular: it has no inverse, its rows are linearly dependent, and the linear system Ax = b has either no solution or infinitely many. Geometrically it flattens space into a lower dimension.
Can a non-square matrix have a determinant?+
No. The determinant is defined only for square (n×n) matrices. For rectangular matrices you can look at the rank, or at determinants of square submatrices.
What happens to the determinant when you swap rows?+
Swapping two rows flips the sign. Multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another leaves it unchanged — these rules are what make elimination work.
What is the determinant of a product or transpose?+
det(AB) = det(A)·det(B) and det(Aᵀ) = det(A). It follows that det(A⁻¹) = 1/det(A) and, for an n×n matrix, det(kA) = kⁿ·det(A).