Question

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

• A. a symmetric matrix
• B. a skew-symmetric matrix
• C. diagonal matrix
• D. does not exist

Answer 1

Answer: (D)

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