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Q.
If $A$ is a $3 \times 3$ skew-symmetric matrix with real entries and trace of $A^2$ equals zero, then
Matrices
Solution:
Let $A=\begin{bmatrix}a & b & c \\ b & d & e \\ c & e & f\end{bmatrix}$
Trace of $A^2=\left(a^2+b^2+c^2\right)+\left(b^2+d^2+e^2\right)
+\left(c^2+e^2+f^2\right) $
$\Rightarrow a^2+2 b^2+2 c^2+d^2+2 e^2+f^2=0$
As $a, b, c, d, e, f$ are real numbers, we get
$a=b=c=d=e=f=0 . $
$\therefore A=0$