Question:
Let $A=\left[a_{i j}\right]$ be a square matrix of order $3 \times 3$ and $C_{i j}$ denote cofactor of $a_{i j}$ in $A$. If $|A|=5$, write the value of $a_{31} C_{31}+a_{32} C_{32} a_{33} C_{33}$.
Solution:
If $A=\left[\mathrm{a}_{\mathrm{i}}\right]$ is a square matrix of order $n$ and $\mathrm{C}_{\mathrm{i} \mathrm{j}}$ is a cofactor of $\mathrm{a}_{\mathrm{i} \mathrm{j}}$, then
$\sum_{i=1}^{n} a_{i j} C_{i j}=|A|$ and $\sum_{j=1}^{n} a_{i j} C_{i j}=|A|$
Given: $|\mathrm{A}|=5$ and matrix $A$ is of order $3 \times 3$
Since $a_{13} C_{13}+a_{23} C_{23}+a_{33} C_{33}$ represent expansion of $A$ along third column, we get
$a_{13} C_{13}+a_{23} C_{23}+a_{33} C_{33}=|A|=5$
$\Rightarrow a_{13} C_{13}+a_{23} C_{23}+a_{33} C_{33}=5$