If $a\ne b\ne c$ all are positive, then the value of determinant $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

Consider the following in respect of two non-singular matrices $A$ and $B$ of same order

$\mathrm{I}$. $det\left(A+B\right)=detA+detB$

$\mathrm{II}$. ${\left(A+B\right)}^{-1}=A^{-1}+B^{-1}$

Which of the above statement $\left(s\right)$ is$/$are correct$?$

If $A=\left[\begin{array}{ll}2& 7\\ 1& 5\end{array}\right]$ then what is $A+3A^{-1}$ equal to$?$ Where, $I$ is the identity matrix of order $2$.

Consider the following statements in respect of the determinant $\left|\begin{array}{cc}{\cos }^2\frac{\alpha }{2}& {\sin }^2\frac{\alpha }{2}\\ {\sin }^2\frac{\beta }{2}& {\cos }^2\frac{\beta }{2}\end{array}\right|$ where $\alpha ,\beta$ are complementary angles.

$\mathrm{I}$. The value of the determinant is $\frac{1}{\sqrt{2}}\cos \left(\frac{\alpha -\beta }{2}\right)$.

$\mathrm{II}$. The maximum value of the determinant is $\frac{1}{\sqrt{2}}$.

Which of the above statement $\left(s\right)$ is $/$ are correct$?$