Compute the cofactor matrix, adjugate (classical adjoint), determinant, and inverse of a 2×2 or 3×3 matrix with step-by-step cofactor breakdowns and visual grids.
The cofactor matrix of a square matrix A is formed by replacing each element aᵢⱼ with its cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ, where Mᵢⱼ is the minor (the determinant of the sub-matrix obtained by deleting row i and column j). The adjugate (classical adjoint) is the transpose of the cofactor matrix.
The adjugate is essential for computing the matrix inverse using the formula A⁻¹ = (1/det(A)) × adj(A). This formula is elegant for theoretical work and practical for 2×2 and 3×3 matrices. It also appears in Cramer's rule for solving systems of linear equations.
This calculator lets you enter a 2×2 or 3×3 matrix and instantly see its cofactor matrix, adjugate, determinant, and inverse (when it exists). Each cofactor is displayed in a grid that shows the sign pattern, minor value, and final cofactor value. The adjugate is presented alongside the original matrix for easy comparison. Bar charts visualize the relative magnitudes of cofactors across the matrix, making patterns immediately apparent.
Cofactor Matrix & Adjugate Calculator helps you solve cofactor matrix & adjugate problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Determinant, Is Invertible?, Cofactor Matrix to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ; adj(A) = [Cᵢⱼ]ᵀ; A⁻¹ = (1/det A) × adj(A)
Result: Determinant shown by the calculator
Using the preset "2 × 2", the calculator evaluates the cofactor matrix & adjugate setup, applies the selected algebra rules, and reports Determinant with supporting checks so you can verify each transformation.
This calculator takes the problem inputs and applies the relevant cofactor matrix & adjugate relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Determinant, Is Invertible?, Cofactor Matrix, Adjugate (adj A) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
The cofactor matrix replaces each element aᵢⱼ with its cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ. It captures each element's contribution to the determinant.
The adjugate is the transpose of the cofactor matrix. It is used in the formula A⁻¹ = (1/det) × adj(A) to compute the matrix inverse.
In classical linear algebra, "adjoint" or "classical adjoint" means the adjugate (transpose of cofactors). In functional analysis, "adjoint" can mean the conjugate transpose. This calculator uses the classical definition.
The inverse does not exist when det(A) = 0 (the matrix is singular). The adjugate still exists, but dividing by zero is undefined.
Multiply A × A⁻¹ — if you get the identity matrix, the inverse is correct. This calculator computes it as (1/det) × adj(A).
Yes, cofactors generalize to any square matrix. However, for n > 4, this approach is impractical. Use Gaussian elimination or LU decomposition instead.