Question

If $\Delta=\begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}$ and $A_{i j}$ is cofactor of $a_{i j}$, then value of $\Delta$ is

• A. $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}$
• B. $a_{11} A_{11}+a_{12} A_{21}+a_{13} A_{31}$
• C. $a_{21} A_{11}+a_{22} A_{12}+a_{23} A_{13}$
• D. $a_{11} A_{11}+a_{21} A_{21}+a_{31} A_{31}$

Answer 1

Answer: (D)

$\Delta$ is equal to the sum of the products of the elements of a row (or a column) with their corresponding cofactors.
$ \therefore \Delta=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \text { or } a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23} $
$ \text { or } a_{31} \Lambda_{31}+a_{32} \Lambda_{32}+a_{33} \Lambda_{33} \text { or } a_{11} \Lambda_{11}+a_{21} \Lambda_{21}+a_{31} \Lambda_{31}$
$\text { or } a_{12} A_{12}+a_{22} A_{22}+a_{32} A_{32} \text { or } a_{13} A_{13}+a_{23} A_{23}+a_{33} A_{33}$
Sum of the products of the elements of first column with their corresponding cofactors is $\Delta=a_{11} A_{11}+a_{21} A_{21}+a_{31} A_{31}$.