# $\text{If } \Delta = \begin{vmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{vmatrix} \text{ and } A_y \text{ is Cofactor of } a_y \text{, then the value of } a_{11}A_{21}+a_{12}A_{22}+a_{13}A_{23}=?$

$\begin{array}{1 1} Hence \bigtriangleup=a_{11}A_{31}+a_{12}A_{32}+a_{13}A_{33} \\Hence \bigtriangleup=a_{11}A_{11}+a_{12}A_{21}+a_{313}A_{31} \\ Hence \bigtriangleup=a_{21}A_{11}+a_{22}A_{21}+a_{23}A_{31} \\ Hence \bigtriangleup=a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31} \end{array}$

Toolbox:
• Determinant ($\bigtriangleup$) of a matrix is the sum of the product of the element of a column (or a row)with its corresponding cofactors.
We know that $\bigtriangleup$=Sum of the product of the element of a column (or a row) with its corresponding cofactors.
But the sum of the product of the elements in any row (or column) with the corresponding cofactors of any other row (or column) is zero.
Hence the correct answer is 0
edited Feb 11, 2014