Determinants

If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero. For example, \(\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}\) But \(a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = a_{11}(-1)^{2+1}|\begin{smallmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{smallmatrix}| + a_{12}(-1)^{2+2}|\begin{smallmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{smallmatrix}| + a_{13}(-1)^{2+3}|\begin{smallmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{smallmatrix}|\) \(= -a_{11}(a_{12}a_{33} - a_{13}a_{32}) + a_{12}(a_{11}a_{33} - a_{13}a_{31}) - a_{13}(a_{11}a_{32} - a_{12}a_{31})\) \(= -a_{11}a_{12}a_{33} + a_{11}a_{13}a_{32} + a_{12}a_{11}a_{33} - a_{12}a_{13}a_{31} - a_{13}a_{11}a_{32} + a_{13}a_{12}a_{31} = 0\) Similarly, we can try for other rows and columns.