Question

If $A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ 1& -\sin \theta & 1\end{array}\right]$ where $0\le \theta \le 2\pi$ then-

• A. Det $\left(A\right)=0$
• B. Det $\left(A\right)\in (0,\infty )$
• C. Det $\left(A\right)\in [0,4]$
• D. Det $\left(A\right)\in [2,\infty )$

Answer 1

Answer: (C)

Det $\left(A\right)\in [0,4]$

$A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ 1& -\sin \theta & 1\end{array}\right]$

$detA=1(1+{\sin }^2\theta )-\sin \theta (-\sin \theta -\sin \theta )+1({\sin }^2\theta -1)$

$=1+{\sin }^2\theta +{\sin }^2\theta +{\sin }^2\theta +{\sin }^2\theta -1=4{\sin }^2\theta$

$\Rightarrow 0\le {\sin }^2\theta \le 1\therefore 0\le 4{\sin }^2\theta \le 4$

$detA\in [0,4]$