If the value of the determinant $\left|\begin{array}{lll}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, where $a\ne b\ne c,$ then the value of $abc$

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$?$

If $A$ is an orthogonal matrix of order $3$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$, then which of the following is/are correct $?$

$\mathrm{I}$. $\left|AB\right|=\pm 47$

$\mathrm{II}$. $AB=BA$

Select the correct answer using the code given below.

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$?$