Minors and cofactors

The minor Mᵢⱼ of the entry aᵢⱼ is the determinant left after deleting row i and column j. In a determinant of order n (n ≥ 2), every minor has order n − 1.

Worked example: in the determinant with rows (1, 2, 3), (4, 5, 6), (7, 8, 9), the entry 6 sits in row 2 and column 3, so M₂₃ = det [[1, 2], [7, 8]] = 8 − 14 = −6.

The cofactor of aᵢⱼ is Aᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ: the minor itself when i + j is even, its negative when i + j is odd.

Worked example: for det [[1, −2], [4, 3]] the minors are M₁₁ = 3, M₁₂ = 4, M₂₁ = −2, M₂₂ = 1, and the cofactors are A₁₁ = 3, A₁₂ = −4, A₂₁ = 2, A₂₂ = 1.

Expansion in cofactor form: ∆ = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃, and the same holds for any row or column: pair each entry with its own cofactor and add.

Pairing the entries of one row with the cofactors of a different row always gives 0, because the sum equals a determinant with two identical rows.

Worked example: for rows (2, −3, 5), (6, 0, 4), (1, 5, −7), the third-row cofactors are A₃₁ = −12, A₃₂ = 22, A₃₃ = 18, and 2 × (−12) − 3 × 22 + 5 × 18 = 0.