Question

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

(a) $\mathrm{Det}\left(A\right)=0$
(b) $\mathrm{Det}\left(A\right)\in \left(2,\infty \right)$
(c) $\mathrm{Det}\left(A\right)\in \left(2,4\right)$
(d) $\mathrm{Det}\left(A\right)\in \left[2,4\right]$

Answer 1

(d) $\mathrm{Det}\left(A\right)\in \left[2,4\right]$

$$\begin{gathered} \left|1\sin \theta 1 \\ -\sin \theta 1\sin \theta \\ -1-\sin \theta 1\right| \\ =\left|1\sin \theta 2 \\ -\sin \theta 10 \\ -1-\sin \theta 0\right|\left[\mathrm{Applying}{\mathrm{C}}_3\to {\mathrm{C}}_3+{\mathrm{C}}_1\right] \\ =2\times \left|-\sin \mathrm{\theta }1 \\ -1-\sin \mathrm{\theta }\right|\left[\mathrm{Expanding}\mathrm{along}{C}_3\right] \\ =2\left({\sin }^2\theta +1\right) \\ \mathrm{Given}:0\le \mathrm{\theta }\le 2\mathrm{\pi } \\ \Rightarrow -1\le \sin \mathrm{\theta }\le 1 \\ \Rightarrow 0\le {\sin }^2\theta \le 1 \\ \left|\mathrm{A}\right|=2\left({\sin }^2\theta +1\right) \\ \left|\mathrm{A}\right|=2\times 1=2\left[\mathrm{\theta }=0\right] \\ =2\times 2=4\left[\mathrm{\theta }=2\mathrm{\pi }\right] \\ \Rightarrow \mathrm{Det}\left(\mathrm{A}\right)\in \left[2,4\right] \end{gathered}$$