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

• 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 $\Delta =a_{11}{A}_{11}+a_{21}{A}_{21}+a_{31}{A}_{31}$.