Question

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

Note: Trace of matrix $A$ denotes the sum of diagonal elements of matrix $A$.

• A. $A=0$
• B. $2A=I$
• C. $A$ is orthogonal
• D. None of these

Answer 1

Answer: (A)

$A=0$

Let $A=\left[\begin{array}{ccc}0& -b& -c\\ b& 0& -a\\ c& a& 0\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{ccc}0& -b& -c\\ b& 0& -a\\ c& a& 0\end{array}\right]\left[\begin{array}{ccc}0& -b& -c\\ b& 0& -a\\ c& a& 0\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{ccc}-b^2-c^2& -ac& ab\\ -ac& -b^2-a^2& -bc\\ ab& -bc& -c^2-a^2\end{array}\right]$

$\therefore tr\left(A^2\right)=-2(a^2+b^2+c^2)=0$

Where $tr\left(A\right)$ is the trace of matrix $A$.

$\Rightarrow a=b=c=0$.

$\therefore A=0$

Hence, option A.