If A is a 3 × 3 skew-symmetric matrix with real entries and trace of A2 equals zero, then

Question

If A is a 3 × 3 skew-symmetric matrix with real entries and trace of A2 equals zero, then (A) A=O (B) 2 A=I (C) A is orthogonal (D) none of these

Tardigrade Admin · Accepted Answer

Let A=[a b c b d e c e f] Trace of A2=(a2+b2+c2)+(b2+d2+e2) +(c2+e2+f2) ⇒ a2+2 b2+2 c2+d2+2 e2+f2=0 As a, b, c, d, e, f are real numbers, we get a=b=c=d=e=f=0 . ∴ A=0

Q. If $A$ is a $3 \times 3$ skew-symmetric matrix with real entries and trace of $A^{2}$ equals zero, then

$A = O$

$2 A = I$

$A$ is orthogonal

none of these

Let $A = ⎣ ⎡ a b c b d e c e f ⎦ ⎤$
Trace of $A^{2} = (a^{2} + b^{2} + c^{2}) + (b^{2} + d^{2} + e^{2}) + (c^{2} + e^{2} + f^{2})$
$\Rightarrow a^{2} + 2 b^{2} + 2 c^{2} + d^{2} + 2 e^{2} + f^{2} = 0$
As $a, b, c, d, e, f$ are real numbers, we get
$a = b = c = d = e = f = 0.$
$∴ A = 0$

Q. If A is a 3×3 skew-symmetric matrix with real entries and trace of A2 equals zero, then

A=O

2A=I

A is orthogonal