Solve the following equations
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,

$\sum_{i} a_{i i}=0$