Question:
If $A=\left[a_{i j}\right]$ is a skew-symmetric matrix, then write the value of $\sum_{i} \sum_{j} a_{i j}$.
Solution:
Given: $A=\left[a_{i j}\right]$ is a skew symmetric matrix.
$\Rightarrow a_{i j}=-a_{j i} \quad$ [For all values of $\left.i, j\right]$
$\Rightarrow a_{i i}=-a_{i i} \quad[$ For all values of $i]$
$\Rightarrow a_{i j}=0$
Now,
$\sum_{i} \sum_{j} a_{i j}=a_{11}+a_{12}+a_{13}+\ldots+a_{21}+a_{22}+a_{23}+\ldots+a_{31}+a_{32}+a_{33}+\ldots$
$=0+a_{12}+a_{13}+\ldots-a_{12}+0+a_{23}+\ldots-a_{13}-a_{23}+0+\ldots$
$=0$
Thus,
$\sum_{i} \sum_{j} a_{i j}=0$