Question

If $A\left(\theta \right)=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 0\end{array}\right]$, then $A(\theta {)}^3$ will be a null matrix if and only if

• A. $\theta =(2k+1)\frac{\pi }{3},k\in I$
• B. $\theta =(4k-1)\frac{\pi }{3}k\in I$
• C. $\theta =(3k-1)\frac{\pi }{4},k\in I$
• D. none of these

Answer 1

The correct option is D none of these

$A\left(\theta \right)=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 0\end{array}\right]$

$\Rightarrow A(\theta {)}^3=\left[\begin{array}{ccc}{\cos }^3\theta & -{\sin }^3\theta & 0\\ {\sin }^3\theta & {\cos }^3\theta & 0\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}{\cos }^3\theta & -{\sin }^3\theta & 0\\ {\sin }^3\theta & {\cos }^3\theta & 0\\ 0& 0& 0\end{array}\right]$

$A(\theta {)}^3$ is null matrix when

${\cos }^3\theta =0,{\sin }^3\theta =0$