Question:
If $A=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ and $C_{i j}$ is cofactor of $a_{i j}$ in $A$, then value of $|A|$ is given by
(a) $a_{11} C_{31}+a_{12} C_{32}+a_{13} C_{33}$
(b) $a_{11} C_{11}+a_{12} C_{21}+a_{13} C_{31}$
(c) $a_{21} C_{11}+a_{22} C_{12}+a_{23} C_{13}$
(d) $a_{11} C_{11}+a_{21} C_{21}+a_{13} C_{31}$
Solution:
(d) $a_{11} C_{11}+a_{21} C_{21}+a_{13} C_{31}$
Properties of determinants state that if $A$ is a square matrix of the order $n$, then Det $(A)$ is the sum of products of elements of a row (or a column) with the corresponding cofactor of that element.
$|A|=a_{11} C_{11}+a_{21} C_{21}+a_{31}, C_{31} \quad$ [Calculating along $\left.C_{1}\right]$