Linear Algebra/Cofactors and Minors

Cofactors

Consider any column, the k^th one. Consider all terms that contain the element a_ik, and factor out a_ik. The sum of all such terms is called the cofactor of a_ik, do be denoted Co(a_ik).

Every term contains exactly one element from the k^th column. A determinant D can thus be written as a_1kCo(a_1k)+a_2kCo(a_2k)+a_3kCo(a_3k)+...a_nkCo(a_nk).

If, in fact the cofactors of a column or row were to be multiplied by some different column and row, its sum would be zero because it would be the same as a determinant with repeat columns and rows.