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. $a_{11}{A}_{11}+a_{12}{A}_{21}+a_{13}{A}_{31}$
• B. $a_{21}{A}_{11}+a_{22}{A}_{12}+a_{23}{A}_{13}$
• C. $a_{11}{A}_{31}+a_{12}{A}_{32}+a_{13}{A}_{33}$
• D. $a_{11}{A}_{11}+a_{21}{A}_{21}+a_1{A}_{31}$

Answer 1

The correct option is D $a_{11}{A}_{11}+a_{21}{A}_{21}+a_1{A}_{31}$

Given : $\Delta =\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$\Delta =$ Sum of products of elements of row (or column) with their corresponding cofactors.
$\Delta =a_{11}{A}_{11}+a_{21}{A}_{21}+a_{31}{A}_{31}$
So, option $D$ is correct.