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  4. If A is a skew-symmetric matrix of odd order n, then

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Q. If $A$ is a skew-symmetric matrix of odd order $n$, then

Determinants

Solution:

Since, A is a skew-symmetric matrix.
$\therefore A^{\prime}=-A$
$\Rightarrow \left|A^{\prime}\right|=|-A|$
$\Rightarrow \left|A^{\prime}\right|=(-1)^n|A|$
$\left(\because|k A|=k^n|A|\right.$, if $A$ is order $\left.n\right)$
$\Rightarrow |A|=(-1)^n|A| \left(\because|A|=\left|A^{\prime}\right|\right)$
$\Rightarrow |A|=-|A| (\because n$ is odd $)$
$\Rightarrow 2|A|=0$
$|A|=0$