If $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right],$ then prove that $\mathrm{A}^{n}=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right], n \in \mathrm{N}$

Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix, where

$a_{i j}= 1 , \quad\quad\text { if } i=j$

$\quad\quad-x ,\quad \text { if }|i-j|=1$

$\quad\quad2 x+1, \text { otherwise }$

Let a function f: $\mathrm{R} \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\operatorname{det}(\mathrm{A})$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:

For how many value $(s)$ of $x$ in the closed interval $[ – 4,\,\, – 1]$ is the matrix $\left[ {\begin{array}{*{20}{c}}3&{ – 1 + x}&2\\3&{ – 1}&{x + 2}\\{x + 3}&{ – 1}&2\end{array}} \right]$  singular