Cofactor & Adjugate Matrix Calculator
Compute the cofactor matrix, the adjugate (adjoint), and the determinant of a 2×2 or 3×3 square matrix.
Invertible when det(A) is nonzero
- 1
Cofactor C₁₁
4 × 6 − 5 × 0 = 24 - 2
Cofactor C₁₂ (sign −)
−(0 × 6 − 5 × 1) = 5 - 3
Cofactor C₁₃
0 × 0 − 4 × 1 = -4 - 4
Determinant (first-row expansion)
1 × 24 + 2 × 5 + 3 × -4 = 22det(A) = A₁₁·C₁₁ + A₁₂·C₁₂ + A₁₃·C₁₃
How does this calculator work?
For a 2×2 or 3×3 matrix, each cofactor Cᵢⱼ equals (-1)^(i+j) times the minor (the determinant left after deleting row i and column j). The cofactor matrix gathers all Cᵢⱼ; the adjugate is its transpose, adj(A) = Cᵀ. When det(A) ≠ 0, the inverse equals adj(A)/det(A).
Formula
How this is calculated
Enter a square matrix A with one row per line, using spaces or commas to separate the entries. The calculator accepts 2×2 and 3×3 matrices; every row must have the same length as the number of rows.
For each position (i, j) it deletes row i and column j to form a smaller submatrix, then takes its determinant — this is the minor Mᵢⱼ. The cofactor is Cᵢⱼ = (-1)^(i+j)·Mᵢⱼ, so the sign alternates in a checkerboard pattern starting with + in the top-left corner. Collecting every Cᵢⱼ gives the cofactor matrix C.
The adjugate (also called the adjoint), adj(A), is the transpose of the cofactor matrix: adj(A) = Cᵀ. The determinant is found by expanding along the first row, det(A) = Σⱼ A₀ⱼ·C₀ⱼ. When det(A) ≠ 0 the inverse is A⁻¹ = adj(A)/det(A). All entries are unitless; if det(A) = 0 the matrix is singular and has no inverse, though the cofactor and adjugate still exist.
Frequently asked questions
The cofactor matrix collects each signed minor Cᵢⱼ in its original position. The adjugate (adjoint) is simply its transpose, adj(A) = Cᵀ, so rows and columns are swapped.
For any invertible matrix, A⁻¹ = adj(A) / det(A). The adjugate gives the numerators and the determinant is the common denominator.
This tool supports 2×2 and 3×3 square matrices only. Non-square input or rows of unequal length will not produce a result.
Also known as
TG we-Calculate Editorial Team. (2026). Cofactor & Adjugate Matrix Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/matrix-cofactor-adjugate-calculator
TG we-Calculate Editorial Team. "Cofactor & Adjugate Matrix Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/matrix-cofactor-adjugate-calculator.
TG we-Calculate Editorial Team, "Cofactor & Adjugate Matrix Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/matrix-cofactor-adjugate-calculator
@misc{wecalculate_matrix_cofactor_adjugate_calculator, title = {Cofactor & Adjugate Matrix Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/matrix-cofactor-adjugate-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?