Question:
In question 18, write the value of $a_{11} C_{21}+a_{12} C_{22}+a_{13} C_{23}$.
Solution:
We know that in a square matrix of order n, the sum of the products of elements of a row (or a column) with the cofactors of the corresponding elements of some other row (or column ) is zero. Therefore,
$A=\left[a_{i j}\right]$ is a square matrix of order $n$.
$\Rightarrow \sum_{j=1}^{n} a_{i j} C_{k j}=0$ and $\sum_{i=1}^{n} a_{i j} C_{i k}=0$
$\Rightarrow a_{11} C_{21}+a_{12} C_{22}+a_{13} C_{23}=0 \quad[$ Since the elements are of first row and the cofactors are of elements of second row $]$
$\Rightarrow \mathrm{a}_{11} \mathrm{C}_{21}+\mathrm{a}_{12} \mathrm{C}_{22}+\mathrm{a}_{13} \mathrm{C}_{23}=0$