Question

Let the matrix Let the matrix $ M=\left[\begin{array}{ccc} \tan \left(\frac{301 \pi}{3}\right) & \sec (2016 \pi) & \cot \left(\frac{2015 \pi}{2}\right) \\ \cot \left(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}\right) & 2 \sin \left(\frac{4 \pi}{3}\right) & \sec ^{-1}(2016)-\cos ^{-1}\left(\frac{1}{2016}\right) \\ \cos \left(\frac{2009 \pi}{2}\right) & \cos ^{-1}(1) & \sec \left(\frac{301 \pi}{3}\right) \end{array}\right] \text {, } $
then the value of $\operatorname{det}\left(2 M^{T}+\operatorname{adj}(M)\right)$ is:

Answer 1

$M=\left[\begin{array}{ccc}\sqrt{3} & 1 & 0 \\ 1 & -\sqrt{3} & 0 \\ 0 & 0 & 2\end{array}\right]$
As, $M M^{T} = 4 I \Rightarrow 2 M^{T} + a d j (M) = 0$
$\Rightarrow \left|\right. 2 M^{T} + a d j (M) \left|\right. = 0$