Question

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

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

Answer 1

The correct option is D $Det\left(A\right)\in [2,4]$

On applying $R_1=R_1+R_3$ to the given matrix, we have
$\left|\begin{array}{ccc}0& 0& 2\\ -\sin \theta & 1& -\sin \theta \\ -1& -\sin \theta & 1\end{array}\right|$
Now on expanding, we have
$2({\sin }^2\theta +1)$
Maximum value of ${\sin }^2\theta =1$
Minimum value of ${\sin }^2\theta =0$
$\therefore Det\left(A\right)\in [2,4]$