Let  $A\left(t\right)=\left[a_{ij}\right]$ is a matrix of order  $3\times 3$  given by $a_{ij}=\{\begin{array}{l}2\cos tifi=j\\ 1if|i-j|=1,then\\ 0otherwise\end{array}$

	Column_I		Column_II
A)	The number of t in interval $[-2\pi ,4\pi ]$ such that $\left|A\left(t\right)\right|=4$ is equal to	P)	0
B)	$\left|A\left(\frac{\pi }{17}\right)\right|\left|A\left(\frac{4\pi }{17}\right)\right|$ is equal to	Q)	1
C)	The maximum value of $\left|A\left(t\right)\right|+\left|A\left(2t\right)\right|,\forall t\in R$  is equal to	R)	4
D)	$\underset{0}{\overset{\pi }{\int }}\left|A\left(t\right)\right|\left|A\left(4t\right)\right|dt$  is equal to	S)	6
		T)	8

$$\begin{gathered} A\left(t\right)=\left[\begin{array}{l}2\cos t10\\ 12\cos t1\\ 012\cos t\end{array}\right] \\ \left|A\left(t\right)\right|=2\cos t(4{\cos }^{2}t-1)-2\cos t=8{\cos }^{3}t-4\cos t \\ \left|A\left(t\right)\right|=4\cos t\cos 2t \end{gathered}$$

$$\begin{gathered} A) \left|A\left(t\right)\right|=4\Rightarrow t=-2n\pi ,n\in 1 \\ \Rightarrow t=-2\pi ,0,2\pi ,4\pi \end{gathered}$$

$$\begin{gathered} B) \left|A\left(\frac{\pi }{17}\right)\right|\left|A\left(\frac{4\pi }{17}\right)\right|=\left|16\cos \frac{\pi }{17}\cos \frac{2\pi }{17}\cos \frac{4\pi }{17}\cos \frac{8\pi }{17}\right|=\left|\frac{\sin \frac{16\pi }{17}}{\sin \frac{\pi }{17}}\right|=1 \\ C) \left|A\left(t\right)\right|+\left|A\left(2t\right)\right|=4\cos t\cos 2t+4\cos 2t\cos 4t\le 8 \\ D) \underset{0}{\overset{\pi }{\int }}16\cos t\cos 2t\cos 4t\cos 8tdt=\underset{0}{\overset{\pi }{\int }}\frac{\sin 16tdt}{\sin t} \\ =\underset{0}{\overset{\pi }{\int }}(\frac{\sin 16t}{\sin t}+\frac{\sin (16\pi -16t)}{\sin (\pi -t)})dt \\ =0 \end{gathered}$$