If $A$ is a square matrix of order $n$ and $A = kB$, where $k$ is a scalar, then $|A|=$

Let $S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}$, where $i=\sqrt{-1} .$ Then the number of $2 -$ digit numbers in the set $\mathrm{S}$ is $……$

Show that

$\left[ {\begin{array}{*{20}{l}}
  1&2&3 \\ 
  0&1&0 \\ 
  1&1&0 
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  { – 1}&1&0 \\ 
  0&{ – 1}&1 \\ 
  2&3&4 
\end{array}} \right]$ $ \ne \left[ {\begin{array}{*{20}{c}}
  { – 1}&1&0 \\ 
  0&{ – 1}&1 \\ 
  2&3&4 
\end{array}} \right]\left[ {\begin{array}{*{20}{l}}
  1&2&3 \\ 
  0&1&0 \\ 
  1&1&0 
\end{array}} \right]$

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P=\left[p_j\right]$ be a $n \times n$ matrix with $p_{i j}=\omega^{i+j}$. Then $P ^2 \neq 0$, when $n =$

$(A)$ $57$ $(B)$ $55$ $(C)$ $58$ $(D)$ $56$

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,