Question

If $\Delta \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$ and $A_{ij}$ is cofactor of $a_{ij}$, then value of $\Delta$ is given by

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

$\Delta$ is equal to the sum of the products of the elements of a row (or a column) with their corresponding cofactors.

$\begin{array}{ccc}\Delta =a_{11}{A}_{11}+a_{12}{A}_{12}+a_{13}{A}_{13}& or& a_{21}{A}_{21}+a_{22}{A}_{22}+a_{23}{A}_{23}\\ or{a}_{31}{A}_{31}+a_{32}{A}_{32}+a_{33}{A}_{33}& or& a_{11}{A}_{11}+a_{21}{A}_{21}+a_{31}{A}_{31}\\ or{a}_{12}{A}_{12}+a_{22}{A}_{22}+a_{32}{A}_{32}& or& a_{13}{A}_{13}+a_{23}{A}_{23}+a_{33}{A}_{33}\end{array}$
Sum of the products of the elements of first column with their corresponding cofactors is $\Delta =a_{11}{A}_{11}+a_{21}{A}_{21}+a_{31}{A}_{31}$
Hence, correct option is (d).