Cofactor Expansion Calculator — 3×3 Determinant
Enter all nine entries of a 3×3 matrix to compute its determinant by cofactor (Laplace) expansion along the first row. Every minor, cofactor and the final sum is shown step by step.
Determinant found by cofactor expansion along row 1
Minor M₁₁ = det([[e,f],[h,k]])
Cofactor C₁₁ = (−1)^(1+1) × M₁₁ = +M₁₁
Minor M₁₂ = det([[d,f],[g,k]])
Cofactor C₁₂ = (−1)^(1+2) × M₁₂ = −M₁₂
Minor M₁₃ = det([[d,e],[g,h]])
Cofactor C₁₃ = (−1)^(1+3) × M₁₃ = +M₁₃
Expand along row 1: det = a·C₁₁ + b·C₁₂ + c·C₁₃
- 1
Minor M₁₁ = ek − fh
(5)(0) − (6)(8) = -48 - 2
Minor M₁₂ = dk − fg
(4)(0) − (6)(7) = -42 - 3
Minor M₁₃ = dh − eg
(4)(8) − (5)(7) = -3 - 4
det A = a·C₁₁ + b·C₁₂ + c·C₁₃
(1)(-48) + (2)(42) + (3)(-3) = 27C₁₂ = −M₁₂ carries sign (−1)^(1+2) = −1.
How does this calculator work?
det(A) = a·C₁₁ + b·C₁₂ + c·C₁₃. Each cofactor Cᵢⱼ = (−1)^(i+j) × det(2×2 sub-matrix with row 1 and column j removed). The sign pattern for row 1 is +, −, +. For the default matrix [1,2,3; 4,5,6; 7,8,0] the determinant is 27. A determinant of zero means the matrix is singular.
Formula
How this is calculated
Cofactor expansion (also called Laplace expansion) expresses the determinant of an n×n matrix as a weighted sum of (n−1)×n−1 determinants. For a 3×3 matrix, expanding along row 1 means multiplying each element aᵢ in that row by its cofactor Cᵢ — the signed 2×2 determinant obtained by crossing out row 1 and column i — then summing the three products.
The sign of each cofactor follows the checkerboard pattern (−1)^(i+j): position (1,1) is positive, (1,2) is negative, and (1,3) is positive. Each 2×2 minor is evaluated with the standard formula det([[p,q],[r,s]]) = ps − qr. The result is the same regardless of which row or column you expand along.
Cofactor expansion is conceptually clear for small matrices but is computationally expensive for large ones — its cost is O(n!) recursive calls. For matrices larger than 4×4, numerical algorithms like LU decomposition are used in practice. The calculator also shows the GridTransform of the top-left 2×2 sub-matrix, whose determinant visually represents how the linear transformation scales area.
Frequently asked questions
Yes. The determinant is the same whether you expand along any row or any column. Choosing a row or column with many zeros reduces arithmetic — for example, a row with two zeros means only one 2×2 determinant needs to be computed.
A zero determinant means the matrix is singular (non-invertible). The rows (or columns) are linearly dependent — at least one row can be expressed as a linear combination of the others — and the system of linear equations A·x = b has either no solution or infinitely many.
The inverse of a matrix is A⁻¹ = (1/det A) × adj(A), where adj(A) is the adjugate (transpose of the cofactor matrix). So cofactor expansion directly produces the ingredients for matrix inversion. If det A = 0, the inverse doesn't exist.
TG we-Calculate Editorial Team. (2026). Cofactor Expansion Calculator — 3×3 Determinant [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cofactor-expansion-old-calculator
TG we-Calculate Editorial Team. "Cofactor Expansion Calculator — 3×3 Determinant." TG we-Calculate. 2026. https://we-calculate.com/calculator/cofactor-expansion-old-calculator.
TG we-Calculate Editorial Team, "Cofactor Expansion Calculator — 3×3 Determinant," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cofactor-expansion-old-calculator
@misc{wecalculate_cofactor_expansion_old_calculator, title = {Cofactor Expansion Calculator — 3×3 Determinant}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cofactor-expansion-old-calculator}}, year = {2026}, note = {TG we-Calculate} }
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