If A = [ cases 1 & sinθ & 1 - sinθ & 1 & sinθ - 1 & - sinθ & 1...

A=\left[$$\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}$$\right]$<br>$\Rightarrow \operatorname{det}(A)=\left(11+\sin ^2 \theta\right)-\sin \theta(-\sin \theta+\sin \theta)+1\left(\sin ^2 \theta+1\right)$<br>$=2+2 \sin ^2 \theta$<br>$\because \theta \in\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)$<br>$\therefore \sin 2 \theta \in\left[0, \frac{1}{2}\right]$<br>$\therefore \operatorname{det}(A) \in[2,3]$which is a subset of \left(\frac{3}{2}, 3\right]$