Cofactor Calculator (3×3 Matrix)

What Is a Cofactor? Definition and Core Formula

A cofactor is a signed version of a matrix minor, and it plays a central role in determinant calculation, matrix inversion, and Cramer's Rule. For any element at row i and column j in a matrix A, the cofactor C_ij is defined by:

C_ij = (−1)^i+j · M_ij

Here, M_ij is the minor — the determinant of the 2×2 submatrix that remains after row i and column j are deleted from the original 3×3 matrix. The factor (−1)^i+j applies the alternating sign that distinguishes a cofactor from its corresponding minor.

Variables Explained

• a₁₁ through a₃₃ — The nine elements of the 3×3 input matrix, arranged in three rows and three columns. Each element occupies a unique (i, j) position.
• i, j — The 1-indexed row and column of the target element. These determine which row and column to delete when forming the 2×2 minor submatrix.
• M_ij — The minor: the determinant of the 2×2 submatrix obtained by removing row i and column j. For a 2×2 matrix [[p, q],[r, s]], the determinant is ps − qr.
• (−1)^i+j — The sign factor. When i + j is even the result is +1; when i + j is odd the result is −1.
• C_ij — The cofactor, the final signed minor at position (i, j).

The 3×3 Cofactor Sign Pattern

For a 3×3 matrix, the alternating sign (−1)^i+j generates the following checkerboard pattern across all nine positions:

• Row 1: C₁₁ = +M₁₁, C₁₂ = −M₁₂, C₁₃ = +M₁₃
• Row 2: C₂₁ = −M₂₁, C₂₂ = +M₂₂, C₂₃ = −M₂₃
• Row 3: C₃₁ = +M₃₁, C₃₂ = −M₃₂, C₃₃ = +M₃₃

Memorizing this pattern avoids re-evaluating the exponent each time and reduces calculation errors.

Step-by-Step Calculation Method

• Step 1 — Identify the target position (i, j). Choose the row and column of the element whose cofactor is needed.
• Step 2 — Delete row i and column j. Cross out the entire i-th row and j-th column from the 3×3 matrix. The four remaining entries form a 2×2 submatrix.
• Step 3 — Compute the minor M_ij. Evaluate the 2×2 determinant using the formula ps − qr, where p, q, r, s are the four remaining entries in reading order.
• Step 4 — Apply the sign factor. Multiply M_ij by (−1)^i+j. If i + j is even, C_ij = M_ij. If i + j is odd, C_ij = −M_ij.

Worked Example

Consider the matrix A = [[3, 1, −2],[0, 4, 5],[−1, 2, 6]]. To find cofactor C₁₂ (row 1, column 2):

• Delete row 1 and column 2. The remaining entries are [[0, 5],[−1, 6]].
• Compute M₁₂ = (0)(6) − (5)(−1) = 0 + 5 = 5.
• Apply the sign: (−1)¹⁺² = −1. Therefore C₁₂ = −1 × 5 = −5.

To find C₂₂ from the same matrix: delete row 2 and column 2, leaving [[3, −2],[−1, 6]]. Minor M₂₂ = (3)(6) − (−2)(−1) = 18 − 2 = 16. Sign: (−1)⁴ = +1. So C₂₂ = 16.

Key Applications of Cofactors

• Determinant via Laplace expansion: The determinant of a 3×3 matrix equals a_i1C_i1 + a_i2C_i2 + a_i3C_i3 along any row i. This result is established formally in DET-0050: The Laplace Expansion Theorem (Ximera/OSU) and derived in detail in Cofactor Expansion and Other Properties of Determinants (Cornell University).
• Adjugate and matrix inverse: The adjugate of A is the transpose of the 3×3 cofactor matrix. The inverse is A⁻¹ = (1/det A) · adj(A), valid when det(A) ≠ 0. This relationship is derived in Cofactor Expansions — Interactive Linear Algebra (Georgia Tech).
• Cramer's Rule: Cofactors underpin the closed-form solution to n×n linear systems, as covered in Determinants and Cramer's Rule (University of Utah).
• Characteristic polynomials: Cofactor expansion of the matrix (A − λI) generates the characteristic polynomial whose roots are the eigenvalues of A.