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 

Let $A = \left[ {\begin{array}{*{20}{c}}1&2\\3&4\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}a&0\\0&b\end{array}} \right]\;,a,b \in N$ then

If $P$ is a $3 \times 3$ real matrix such that $P ^{ T }=a P +( a -1) I$, where $a > 1$, then $……….$

Let $A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$. Let $\alpha, \beta \in R$ be such that $\alpha A^{2}+\beta A=2 I$. Then $\alpha+\beta$ is equal to –

If a matrix has $18$ elements, what are the possible orders it can have? What, if it has $5$ elements ?