A skew-symmetric matrix S satisfies the relation S2+I=0, where I is a unit matrix; then S is

Question

A skew-symmetric matrix S satisfies the relation S2+I=0, where I is a unit matrix; then S is (A) Idempotent (B) Orthogonal (C) Involuntary (D) None of these

Tardigrade Admin · Accepted Answer

As S is skew-symmetric, ST=-S. We have S2+I=O ⇒ S2=-I or S S=-I ⇒ S(-S)=I ⇒ S ST=I

Q. A skew-symmetric matrix $S$ satisfies the relation $S^{2} + I = 0$ , where $I$ is a unit matrix; then $S$ is

Idempotent

Orthogonal

Involuntary

None of these

As $S$ is skew-symmetric, $S^{T} = - S$ .
We have $S^{2} + I = O$
$\Rightarrow S^{2} = - I$
or $SS = - I$
$\Rightarrow S (- S) = I$
$\Rightarrow S S^{T} = I$

Q. A skew-symmetric matrix S satisfies the relation S2+I=0, where I is a unit matrix; then S is