Question

A square matrix $A$ is skew-symmetric, such that $A^2+I=O,$ then

• A. $A$ is orthogonal matrix
• B. $A$ is orthogonal matrix of odd order
• C. $A$ is orthogonal matrix of even order
• D. $A$ is involutry matrix

Answer 1

The correct option is C $A$ is orthogonal matrix of even order

Given $A$ is skew-symmetric matrix and $A^2+I=0$

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

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

$\Rightarrow A\text{is orthogonal}$

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.