Let $\mathrm{E}\left(\mathrm{\alpha }\right)=\left[\begin{array}{ll}{\cos }^2\mathrm{\alpha }& \cos \mathrm{\alpha }\sin \mathrm{\alpha }\\ \cos \mathrm{\alpha }\sin \mathrm{\alpha }& {\sin }^2\mathrm{\alpha }\end{array}\right]$ . If $\mathrm{\alpha }$ and $\mathrm{\beta }$ differs by an odd multiple of $\frac{\mathrm{\pi }}{2}$, then $\mathrm{E}\left(\mathrm{\alpha }\right)\mathrm{E}\left(\mathrm{\beta }\right)$ is a -

If $B$ is a $3\times 3$  matrix such that $B^2=0$, then $det. \left[{\left(I+B\right)}^{50}-50B\right]$ is equal to :

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I$ , where $I$is $3\times 3$ identity matrix, then the ordered pair $\left(a, b\right)$ is equal to