Question

If elements of a row (or column) are multiplied with cofactors of any other row (or column). Then their sum is zero.

Answer 1

Multiplying a row by the co-factors of any other row will mean that the row itself is duplicated in the determinant being evaluated. It is then like calculating a determinant with two equal rows. And we know that a determinant with elementary row operations is the same determinant (value wise). So a determinant with two identical rows will be a determinant with a row replaced by difference of those rows (a row full of zeros) and thus it will be zero.