Cofactor Matrix Calculator

Enter a 3×3 matrix to compute its full cofactor matrix (every signed minor Cᵢⱼ), the determinant via cofactor expansion, and the adjugate matrix adj(A) = Cᵀ.

Quick Facts

Cofactor formula
Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ
Mᵢⱼ is the minor: the determinant left after deleting row i and column j.
Determinant link
det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃
Expanding along any row or column of A gives the same determinant.

Your Results

Calculated
Cofactor matrix C(A)
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Every signed minor Cᵢⱼ
Determinant det(A)
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Via cofactor expansion, row 1
Adjugate matrix adj(A)
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Transpose of the cofactor matrix
Classification
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Invertible or singular

Ready

Enter a 3×3 matrix and press Calculate.

How the Cofactor Matrix Calculator works

This tool computes the full cofactor matrix of a 3×3 matrix A, along with the determinant (found by cofactor expansion) and the adjugate matrix. The cofactor matrix is the foundation of two classical results in linear algebra: the cofactor-expansion formula for the determinant, and the adjugate-based formula for the matrix inverse.

Formula and method

For each entry aij of A, first form the minor Mij — the determinant of the 2×2 matrix left after deleting row i and column j. The cofactor attaches a sign based on position:

Cij = (−1)i+j × Mij

The signs form a checkerboard pattern starting with + in the top-left corner: +, −, +, −, +, −, +, −, +. Collecting all nine cofactors gives the cofactor matrix C(A). Two further results follow directly from it: the determinant is a weighted sum of any row's entries and their cofactors — det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ when expanding along row 1 — and the adjugate adj(A) is simply the transpose of C(A), which in turn gives the inverse: A⁻¹ = adj(A) / det(A) whenever det(A) ≠ 0.

Common sources of error

  • Missing the sign: forgetting the (−1)i+j factor turns a cofactor into a plain minor — always check the checkerboard sign pattern for the entry's row and column.
  • Mixing up rows and columns: Mij deletes row i and column j, not row j and column i — a swapped index produces the transpose of the correct minor.
  • Confusing cofactor matrix with adjugate: the adjugate is the transpose of the cofactor matrix, C(A)ᵀ, not C(A) itself — they only coincide when C(A) is symmetric.

Checking your result

Cofactor expansion works along any row or column, so pick a second one and confirm you get the same determinant — for example, expand along column 1 (a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁) and check it matches the row-1 expansion. If det(A) ≠ 0, you can also verify A × adj(A) equals det(A) times the identity matrix.

Applications

The cofactor matrix underlies Cramer's rule for solving linear systems, the adjugate-based inverse formula (useful for exact, fraction-free computation before dividing by the determinant), and symbolic computer algebra systems that need an inverse without introducing rounding error early. It generalizes the same idea used for 2×2 matrices, where the cofactor matrix reduces to just swapping and negating two entries.

Frequently Asked Questions

What is a cofactor of a matrix?
The cofactor Cij of a square matrix A is a signed minor: Cij = (−1)i+j × Mij, where Mij is the determinant of the smaller matrix left after deleting row i and column j from A. The sign alternates in a checkerboard pattern starting with + in the top-left position.
How is the cofactor matrix used to find the determinant?
Cofactor expansion along any row or column gives the determinant. Expanding along row 1 of a 3×3 matrix: det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃. Expanding along any other row or column produces the same value, which is a useful way to check your work.
What is the difference between a minor, a cofactor, and the adjugate?
A minor Mij is the determinant of the submatrix formed by deleting row i and column j. A cofactor Cij applies a sign, (−1)i+j, to that minor. The adjugate (or adjoint) is the transpose of the full matrix of cofactors, and dividing it by the determinant gives the inverse: A⁻¹ = adj(A) / det(A).
Does every matrix have a cofactor matrix?
Yes, as long as the matrix is square. The cofactor matrix and adjugate are defined even when the determinant is zero — only the inverse (adj(A) divided by det(A)) fails to exist for a singular matrix, since division by zero is undefined.

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