Cofactor Matrix Calculator

The cofactor matrix of a square matrix is a matrix where each element C_ij is the cofactor of the corresponding element in the original matrix. Each cofactor is computed as C_ij = (−1)^i+j × M_ij, where M_ij is the minor — the determinant of the submatrix formed by removing row i and column j.

For a 2×2 matrix [[a, b], [c, d]], the four cofactors are: C₁₁ = d, C₁₂ = −c, C₂₁ = −b, C₂₂ = a. Each minor of a 2×2 matrix is just a single element (after removing one row and one column), and the sign factor (−1)^i+j alternates in a checkerboard pattern.

A minor M_ij is the determinant of the submatrix obtained by deleting row i and column j from the original matrix. A cofactor C_ij is the signed minor: C_ij = (−1)^i+j × M_ij. The sign factor creates a checkerboard pattern of +1 and −1 values across the matrix.

To find the inverse of a matrix using cofactors: (1) compute the cofactor matrix, (2) transpose it to get the adjugate matrix, and (3) divide every element by the determinant of the original matrix. The formula is: A⁻¹ = (1 / det(A)) × adj(A). This method only works when det(A) ≠ 0.

The adjugate matrix (also called the classical adjoint) is the transpose of the cofactor matrix. If C is the cofactor matrix of A, then adj(A) = C^T. The adjugate is used to compute the matrix inverse via A⁻¹ = adj(A) / det(A), and it satisfies the identity A × adj(A) = det(A) × I.

Yes, every square matrix has a cofactor matrix, regardless of whether it is invertible. However, the cofactor matrix is most useful when computing the inverse, which requires a non-zero determinant. If det(A) = 0, the matrix is singular and not invertible, but the cofactor matrix still exists.

The determinant of a matrix can be computed by expanding along any row or column using cofactors — this is known as cofactor expansion (Laplace expansion). For row i: det(A) = Σ a_ij × C_ij for all j. This recursive relationship means minors and cofactors are fundamental building blocks for determinant computation.

This cofactor matrix calculator supports 2×2, 3×3, and 4×4 square matrices. Select your desired size from the dropdown menu, fill in all matrix elements, and the cofactor matrix, determinant, and adjugate matrix are computed for you automatically.