Question

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

• A. $|A|=0$
• B. $|A|=-1$
• C. $|A|=\left|A^{\prime }\right|$
• D. None of these

Answer 1

Answer: (A)

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