Assume $X,\, Y,\, Z, W$ and $P$ are the matrices of order $2 \times n, \,3 \times k,\, 2 \times p, \,n \times 3$ and $p \times k$ respectively. The restriction on $n,\, k$ and $p$ so that $P Y+W Y$ will be defined are :

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 $a$ , $b$ , $c$ are non real numbers satisfying equation $x^5 = 1$ and $S$ be the set of all non-invertible matrices of the form $\left[ {\begin{array}{*{20}{c}}
  1&a&b \\ 
  w&1&c \\ 
  {{w^2}}&w&1 
\end{array}} \right],\,\,\,\,w = {e^{\frac{{i\,2\pi }}{5}}}$ . Then the number of distinct matrices in the set $S$ is 

For the matrix $A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]$. find the number $a$ and $b$ such that $A^{2}+a A+b I=0$