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  4. The inverse of a skew-symmetric matrix of odd order is

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Q. The inverse of a skew-symmetric matrix of odd order is

Matrices

Solution:

Let $A$ be a skew-symmetric, matrix of order $n$. By definition
$A^{\prime}=-A $
$\Rightarrow \left|A^{\prime}\right|=|-A| \Rightarrow|A|=(-1)^n|A|$
$\Rightarrow |A|=-|A| [\because n \text { is odd }]$
$\Rightarrow 2|A|=0 \Rightarrow |A|=0$
$\therefore A^{-1} \text { does not exist. }$