Question

A skew-symmetric matrix $A$ is such that $A^2+I=O,$ then which of the following is/are correct

• A. $A$ is orthogonal matrix.
• B. Order of $A$ is even.
• C. Order of $A$ is odd.
• D. $A$ is involutory matrix.

Answer 1

Answer: (A), (B)

Order of $A$ is even.

Given : $A$ is skew-symmetric matrix and $A^2+I=O$

$i.e{A}^{\prime }=-A$

$I=-A^2=(-A)A=A^{\prime }A$

$\Rightarrow A\text{is orthogonal}$

Now, $A^{\prime }A=I_n\Rightarrow |A^{\prime }A|=1$

$\Rightarrow |A^{\prime }||A|=1\Rightarrow (-1{)}^n(|A|{)}^2=1$
$(\because (|-A|)=(-1{)}^n|A|)$

$\therefore A$ is an orthogonal matrix of even order.