If $\left[\begin{array}{ccc}x+3 & z+4 & 2 y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0\end{array}\right]=\left[\begin{array}{ccc}0 & 6 & 3 y-2 \\ -6 & -3 & 2 c+2 \\ 2 b+4 & -21 & 0\end{array}\right]$ then find the values of $a,\, b, \,c, \,x, \,y$ and $z$.

$A$  and $B$ are two non- singular square matrices of each $3 \times 3$ such that $AB = A$ and $\left| {A + B} \right| \ne 0$, then

If $A = \left[ {\begin{array}{*{20}{c}}1&1\\0&1\end{array}} \right]$, then ${A^n} = $

Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]$, where $B_1$, $\mathrm{B}_2, \mathrm{~B}_3$ are column matrices, and $\mathrm{AB}_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$, $\mathrm{AB}_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right], \mathrm{AB}_3=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$ If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3+\beta^3$ is equal to

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 $\frac{\pi }{2}$.