Question

The sum of product of elements of any row or column with corresponding co-factors of some other row column is zero.

Answer 1

it will be zero

multiplying a row by the cofactors of any other row will mean that the row itself is duplicated on the determinant being evaluated. It is like calculating a determinant with two equal rows. And we know that a determinant with elementry row operations is the same determinant. So a determinant with two identical rows will be a determinant with a row replaced by different of those rows. Thus it will be zero.