Question:
If $A=\left[a_{i j}\right]$ is a skew-symmetric matrix, then write the value of $\sum_{i} a_{i j}$.
Solution:
Given : $A=\left[a_{i j}\right]$ is a skew - symmetric matrix.
$\Rightarrow a_{i j}=-a_{i j} \quad$ [For all values of $\left.i, j\right]$
$\Rightarrow a_{i i}=-a_{i i} \quad$ [For all values of $i$ ]
$\Rightarrow a_{i j}+a_{i i}=0$
$\Rightarrow 2 a_{i i}=0$
$\Rightarrow a_{i i}=0$ [For all values of $i$ ]
$\sum_{i} a_{i i}=0+0+\ldots+0 \quad[$ itimes $]$
$=0$
Thus,
$\sum_{i} a_{i i}=0$
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