The discussion explains that the sum of the elements of a matrix in one row multiplied by the cofactors from a different row equals zero due to properties of determinants. When a row in a square matrix is replaced with another, the determinant remains unchanged, leading to the conclusion that the expansion using elements from one row and cofactors from another results in a determinant that cancels out. This is because the cofactors are defined in relation to their respective rows, and mixing them leads to a zero result. The discussion emphasizes the consistency of determinant properties across row operations. Ultimately, the mathematical reasoning confirms that the determinant's value is invariant under such transformations.