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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,

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

 

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