Solve this

Solution:

(d) $\operatorname{Det}(A) \in[2,4]$

$\mid \begin{array}{lll}1 & \sin \theta & 1\end{array}$

$-\sin \theta \quad 1 \quad \sin \theta$

$\begin{array}{lll}-1 & -\sin \theta & 1 \mid\end{array}$

$=\mid \begin{array}{lll}1 & \sin \theta & 2\end{array}$

$\begin{array}{lll}-\sin \theta & 1 & 0\end{array}$

$=2\left(\sin ^{2} \theta+1\right)$

Given : $0 \leq \theta \leq 2 \pi$

$\Rightarrow-1 \leq \sin \theta \leq 1$

$\Rightarrow 0 \leq \sin ^{2} \theta \leq 1$

$|\mathrm{A}|=2\left(\sin ^{2} \theta+1\right)$

$|\mathrm{A}|=2 \times 1=2 \quad[\theta=0]$

$=2 \times 2=4 \quad[\theta=2 \pi]$

$\Rightarrow \operatorname{Det}(\mathrm{A}) \in[2,4]$