Let $A$ and $B$ be any two $n \times n$ matrices such that the following conditions hold: $A B=B A$ and there exist positive integers $k$ and $l$ such that $A^k=I$ ( the identity matrix) and $B^l=0$ (the zero matrix). Then,

If $A =$ $\left[ {\,\begin{array}{*{20}{c}}{{{\cos }^2}\alpha }&{\sin \alpha \,\,\cos \alpha }\\{\sin \alpha \,\,\cos \alpha }&{{{\sin }^2}\alpha }\end{array}\,} \right]$; $B =$ $\left[ {\,\begin{array}{*{20}{c}}{{{\cos }^2}\beta }&{\sin \beta \,\,\cos \beta }\\{\sin \beta \,\,\cos \beta }&{{{\sin }^2}\beta }\end{array}\,} \right]$ are such that, $AB$ is a null matrix, then which of the following should necessarily be an odd integral multiple of $\frac{\pi }{2}$.

Let $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in R$ and $A^{4}=\left[a_{i j}\right] .$ If $a_{11}=109,$ then $a_{22}$ is equal to

The value of $\theta$ lying between $- \,\frac{\pi }{4}$ and $\frac{\pi }{2}$ and $0 \le A \le \frac{\pi }{2}$ and satisfying the equation $\left| {\begin{array}{*{20}{c}}{1\, + \,{{\sin }^2}A}&{{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{1\, + \,{{\cos }^2}A}&{2\,\sin \,4\theta }\\{{{\sin }^2}A}&{{{\cos }^2}A}&{1\, + \,2\,\sin \,4\theta }\end{array}} \right|$ $= 0$ are :