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 $\det \left(2M^T+\mathrm{adj}(M)\right)$ is:

Answer 1

Answer: 0

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