Why Does Sum of Matrix Elements x Cofactor in Different Rows Equal Zero?

[ajayguhan]

In a matrix the sum of element of a matrix in a row times it's co factor of that elemt gives the determinant value, but why does the sum of element of a matrix times cofactor of different row is always zero?

 

[Erland]

In a square matrix A, replace row j by row i, obtaining a new matrix B with row i and row j being equal. Expanding using elements from row i in A and cofactors corresponding to row j in A will be the same as expanding det B along row j (using cofactors corresponding to row j in B also) since the elements and cofactors are the same in both cases. So the result is det B, which is...?