Linear Algebra/Cofactors and Minors
Consider any column, the kth one. Consider all terms that contain the element aik, and factor out aik. The sum of all such terms is called the cofactor of aik, do be denoted Co(aik).
Every term contains exactly one element from the kth column. A determinant D can thus be written as a1kCo(a1k)+a2kCo(a2k)+a3kCo(a3k)+...ankCo(ank).
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.