Minors and cofactors

Two names for the pieces used in expanding a determinant.

• Minor Mᵢⱼ: delete row i and column j; the determinant that is left is the minor of the element aᵢⱼ.
• Cofactor Aᵢⱼ = (–1)^(i + j) Mᵢⱼ: the minor with its sign. When i + j is even the sign is +, when odd it is –. This is the same + – + pattern as before.

Example: in |2, 1; 3, 4|:

Two facts to remember, both very common in exams:

• Elements of a row × their own cofactors, added, give the determinant. Row 1: 2 × 4 + 1 × (–3) = 5, and indeed 8 – 3 = 5. The same works for any row or column.
• Elements of one row × the cofactors of a different row, added, always give 0. Row 1 elements with row 2 cofactors: 2 × (–1) + 1 × 2 = 0.

In a short-answer question, you can use these facts without expanding anything, but write the reason in one line, for example: 'Since the elements of the first row are multiplied by the cofactors of the second row, the sum is 0.' Teachers mark the reason, not the exact wording.

For a 3 × 3 determinant, each minor is a 2 × 2 determinant. A sum written with Σ, like Σ aᵢ₂Aᵢ₂, just means: put i = 1, 2, 3 and add. That is column 2 times its own cofactors, so it equals |A|.