Question

A skew symmetric matrix $S$ satisfies the relation $S^2+I=0$ where $I$ is a an unitary matrix, the $S$ is an

• A. idempotent matrix
• B. involuntory matrix
• C. orthogonal matrix
• D. None of these

Answer 1

The correct option is C orthogonal matrix

Given $S$ is a skew-symmetric matrix so,

$S^T=-S\Rightarrow S=-S^T$

(transpose)

$S^2+I=0$

$S^2=-I$

$S.S=-I$

$S.S.S^T=-I-S^T$

Multiplying $64$

$S^T$ on both side $=I(-S^T)$

$=I.S$

$S.S.S^T=S$

multiplying by $S^{-1}$ on both sides

$S^{-1}S.S.S^T=5^{-1}S$

$S.S^T=I$

So, $S$ is an orthogonal matrix